Research ArticleOn Harmonic Functions Defined by Differential Operator withRespect to 119896-Symmetric Points

Afaf A Ali Abubaker and Maslina Darus

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 BangiSelangor D Ehsan Malaysia

Correspondence should be addressed to Maslina Darus maslinaukmedumy

Received 29 April 2014 Revised 4 July 2014 Accepted 7 July 2014 Published 23 July 2014

Academic Editor Heinrich Begehr

Copyright copy 2014 A A A Abubaker and M Darus This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We introduce new classes 119872119867120590119904

119896(120582 120575 120572) and 119872119867

120590119904

119896(120582 120575 120572) of harmonic univalent functions with respect to 119896-symmetric points

defined by differential operator We determine a sufficient coefficient condition representation theorem and distortion theorem

1 Introduction

A continuous function 119891 = 119906 + 119894V is a complex valuedharmonic function in a complex domain119862 if both 119906 and V arereal harmonic in 119862 In any simply connected domain 119863 sub 119862

we can write 119891(119911) = ℎ + 119892 where ℎ and 119892 are analytic in119863 We call ℎ the analytic part and 119892 the coanalytic part of119891 A necessary and sufficient condition for 119891 to be locallyunivalent and sense preserving in 119863 is that |ℎ1015840(119911)| gt |119892

1015840

(119911)|

in 119863 See Clunie and Shell-Small (see [1])Thus for 119891 = ℎ + 119892 isin 119878

lowast

119867 we may write

ℎ (119911) = 119911 +

infin

sum

119899=2

119886119899119911119899

119892 (119911) =

infin

sum

119899=1

119887119899119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1 (1)

Note that 119878lowast119867 reduces to 119878lowast the class of normalized analytic

univalent functions if the coanalytic part of 119891 = ℎ + 119892 isidentically zero Also denote by 119878119867 the subclasses of 119878

lowast

119867

consisting of functions 119891 that map 119880 onto starlike domainA function 119891 is said to be starlike of order 120572 in119880 denoted

by 119878119867(120572) (see [2]) if

120597

120597120579(arg119891 (119903119890

119894120579

)) = Im

(120597120597120579) 119891 (119903119890119894120579

)

119891 (119903119890119894120579)

= R119911ℎ1015840

(119911) minus 1199111198921015840 (119911)

ℎ (119911) + 119892 (119911)

ge 120572

|119911| = 119903 lt 1

(2)

A function 119891 of normalized univalent analytic functions issaid to be starlike with respect to symmetrical points in 119880 ifit satisfies

R1199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911) gt 0 119911 isin 119880 (3)

this class was introduced and studied by Sakaguchi in 1959[3] Some related classes are studied by Shanmugam et al [4]

In 1979 Chand and Singh [5] defined the class of starlikefunctions with respect to 119896-symmetric points of order 120572 (0 le

120572 lt 1) Related classes are also studied by das and Singh[6] Ahuja and Jahangiri [7] discussed the class 119878119867(120572)

which denotes the class of complex-valued sense-preservingharmonic univalent functions119891 of the form (1) and satisfyingthe condition

Im

2 (120597120597120579) 119891 (119903119890119894120579

)

119891 (119903119890119894120579) minus 119891 (minus119903119890119894120579) ge 120572 (4)

In [8] the authors introduced and studied the class 119878119867119896(120572)

which denotes the class of complex-valued sense-preservingharmonic univalent functions 119891 of the form (1) and

ℎ119896(119911) = 119911 +

infin

sum

119899=2

120601119899119886119899119911119899

119892119896(119911) =

infin

sum

119899=1

120601119899119887119899119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1

(5)

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 628972 6 pageshttpdxdoiorg1011552014628972

2 International Journal of Mathematics and Mathematical Sciences

where

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

(119896 ge 1 120576119896

= 1) (6)

From the definition of 120601119899we know

120601119899=

1 119899 = 120580119896 + 1

0 119899 = 120580119896 + 1(119899 ge 2 120580 119896 ge 1) (7)

The differential operator 119863120590119904

120582120575was introduced by Ali Abu-

baker and Darus [9] We define the differential operator ofthe harmonic function 119891 = ℎ + 119892 given by (5) as

119863120590119904

120582120575119891 (119911) = 119863

120590119904

120582120575ℎ (119911) + (minus1)

119904

119863120590119904

120582120575119892 (119911) (8)

where

119863120590119904

120582120575ℎ (119911) = 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 119886

119899119911119899

119863120590119904

120582120575119892 (119911) =

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 119887

119899119911119899

(9)

and also 120595119899(120582 120575 120590 119904) = 119899

119904

(119862(120575 119899)[1 + 120582(119899 minus 1)])120590 120582 ge 0

119862(120575 119899) = (120575 + 1)119899minus1

(119899 minus 1) for 120575 120590 119904 isin 1198730= 0 1 2

and (119909)119899is the Pochhammer symbol defined by

(119909)119899=

Γ (119909 + 119899)

Γ (119909)

= 1 119899 = 0

119909 (119909 + 1) sdot sdot sdot (119909 + 119899 minus 1) 119899 = 1 2 3

(10)

We note that when 119904 = 0 120590 = 1 and 120582 = 0we obtain the Rus-cheweyh derivative for harmonic functions (see [7]) when120590 = 0weobtain the Salagean operator for harmonic functions(see [10]) and when 120590 = 1 119904 = 0 we obtain the operator forharmonic functions given by Al-Shaqsi and Darus [11]

Let 119872119867120590119904

119896(120582 120575 120572) denote the class of complex-valued

sense-preserving harmonic univalent functions 119891 of theform (5) which satisfy the condition

Im

(120597120597120579)119863120590119904

120582120575119891 (119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

= R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572

(11)

where 119911 = 119903119890119894120579 0 le 119903 lt 1 0 le 120579 lt 120587 0 le 120572 lt 1 and the

functions 119863120590119904120582120575

ℎ119896and 119863

120590119904

120582120575119892119896are of the form

119863120590119904

120582120575ℎ119896(119911) = 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899119886119899119911119899

119863120590119904

120582120575119892119896(119911) =

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899119887119899119911119899

(12)

Further denote by 119872119867120590119904

119896(120582 120575 120572) the subclasses of

119872119867120590119904

119896(120582 120575 120572) such that the functions ℎ and 119892 in 119891 = ℎ + 119892

are of the form

ℎ (119911) = 119911 minus

infin

sum

119899=2

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

119892 (119911) =

infin

sum

119899=1

10038161003816100381610038161198871198991003816100381610038161003816 119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1

(13)

and the functions ℎ119896and 119892

119896in 119891119896= ℎ119896+ 119892119896are of the form

ℎ119896(119911) = 119911 minus

infin

sum

119899=2

120601119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

119892119896(119911) =

infin

sum

119899=1

120601119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1

(14)

In this paper we obtain inclusion properties and coefficientconditions for the class 119872119867

120590119904

119896(120582 120575 120572) A representation

theorem and distortion bounds for the class 119872119867120590119904

119896(120582 120575 120572)

are also established

Lemma 1 (see [12]) Let 119891 = ℎ + 119892 isin 119878119867 if

infin

sum

119899=2

1 minus 120572

2 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

infin

sum

119899=1

1 + 120572

2 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816 le 1 (15)

where 0 le 120572 lt 1 then 119891 is harmonic sense-preserving univa-lent in 119880 and 119891 is starlike harmonic of order 120572

2 Main Results

First we give a meaningful conclusion about the class119872119867120590119904

119896(120582 120575 120572)

Theorem 2 Let 119891 isin 119872119867120590119904

119896(120582 120575 120572) where f is given by (1)

then 119891119896defined by (5) is in 119872119867

120590119904

1(120582 120575 120572) = 119872119867

120590119904

(120582 120575 120572)

Proof Let 119891 isin 119872119867120590119904

119896(120582 120575 120572) Then substituting 119903119890

119894120579 by120576]119903119890119894120579 where 120576

119896

= 1 (] = 0 1 119896 minus 1) in (11) respectivelywe have

Im

(120597120597120579)119863120590119904

120582120575119891 (120576

]119903119890119894120579

)

119863120590119904

120582120575119891119896(120576]119903119890119894120579)

ge 120572 (16)

According to the definition of 119891119896and 120576

119896

= 1 we know119891119896(120576

]119903119890119894120579

) = 120576]119891119896(119903119890119894120579

) for any ] = 0 1 119896minus1 and summingup we can get

Im1

119896

119896minus1

sum

]=0

(120597120597120579)119863120590119904

120582120575119891 (120576

]119903119890119894120579

)

120576]119863120590119904

120582120575119891119896(119903119890119894120579)

= Im

(120597120597120579)119863120590119904

120582120575119891119896(119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

ge 120572

(17)

that is 119891119896isin 119872119867

120590119904

(120582 120575 120572)

International Journal of Mathematics and Mathematical Sciences 3

Next a sufficient coefficient condition for harmonic func-tions in 119872119867

120590119904

(120582 120575 120572) is given

Theorem 3 Let 119891 = ℎ + 119892 with ℎ and 119892 given by (1) and119891119896= ℎ119896+ 119892119896with ℎ

119896and 119892

119896given by (5) Let

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904) [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(18)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 Then 119891 is sense-preserving harmonic univalent in119880 and 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

Proof Sinceinfin

sum

119899=1

[119899 minus 120572

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

le

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572120601119899

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1

=

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(19)

by Lemma 1 we conclude that 119891 is sense-preservingharmonic univalent and starlike in 119880 To prove 119891 isin

119872119867120590119904

119896(120582 120575 120572) according to the condition (11) we need to

show that

Im

(120597120597120579)119863120590119904

120582120575119891 (119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

= R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= R119860 (119911)

119861 (119911)ge 120572

119860 (119911) = 119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 119886

119899119911119899

minus (minus1)119904

times

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 119887

119899119911119899

(20)

119861 (119911) = 119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899119886119899119911119899

+ (minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899119887119899119911119899

(21)

where

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1 (22)

Using the fact that R(120596) ge 120572 if and only if |1 minus 120572 + 120596| ge

|1 + 120572 minus 120596| it suffices to show that

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (23)

On the other hand for119860(119911) and 119861(119911) as given in (20) and(21) respectively we have

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

=

10038161003816100381610038161003816100381610038161003816

(1 minus 120572)119863120590119904

120582120575ℎ119896(119911) + 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 minus 120572)119863120590119904

120582120575119892119896(119911) minus 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

minus

10038161003816100381610038161003816100381610038161003816

(1 + 120572)119863120590119904

120582120575ℎ119896(119911) minus 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 + 120572)119863120590119904

120582120575119892119896(119911) + 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(2 minus 120572) 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899] 119886119899119911119899

minus (minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 minus 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

minus 120572119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899] 119886119899119911119899

minus(minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

4 International Journal of Mathematics and Mathematical Sciences

ge (2 minus 120572) |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003817100381710038171003817119887119899

1003817100381710038171003817 |119911|119899

minus 120572 |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899

= (2 minus 120572) |119911| 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1

ge 2 (1 minus 120572) 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 ge 0

(24)

Note that by substituting the value of 120601119899given by (7) in the

previous inequality above then

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

ge 2 (1 minus 120572)

1 minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572 + 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

minus(119899 minus 1) 119896 + 120572 + 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816]

ge 0

(25)

by (18) Thus concludes the proof of the theorem

Next the condition (18) is also necessary for functions in119872119867120590119904

119896(120582 120575 120572) which is clarified

Theorem 4 Let119891 = ℎ+119892with ℎ and 119892 given by (13) and 119891119896=

ℎ119896+119892119896with ℎ

119896and 119892

119896given by (14) Then 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

if and only ifinfin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120588 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(26)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 given by (6)

Proof The if part follows fromTheorem 3 upon noting that ifthe analytic and coanalytic parts of119891 = ℎ+119892 isin 119872119867

120590119904

119896(120582 120575 120572)

are of the form (13) then 119891 isin 119872119867120590119904

119896(120582 120575 120572) For the only if

part we show that 119891 notin 119872119867120590119904

119896(120582 120575 120572) if the condition (26)

does not hold Thus we can write

R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572 (27)

this is equivalent to

0 le R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

minus 120572 (28)

That is

R

((1 minus 120572) 119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119911119899

minus(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119911119899)

times (119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

+(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899)

minus1

ge 0

(29)

The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have

((1 minus 120572) minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119903119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119903119899minus1

)

International Journal of Mathematics and Mathematical Sciences 5

times (1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1

+

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119903119899minus1

)

minus1

ge 0

(30)

If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis

contradicts the required condition for 119891 isin 119872119867120590119904

119896(120582 120575 120572)

and the proof is complete

Now the distortion result is given

Theorem 5 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) minus

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

1003816100381610038161003816119891 (119911)1003816100381610038161003816 le (1 +

100381610038161003816100381611988711003816100381610038161003816) +

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(31)

Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867

120590119904

119896(120582 120575 120572) Taking

the absolute value of 119891(119911) we obtain

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 119903119899

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

times

infin

sum

119899=2

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

2 minus 120572

infin

sum

119899=2

120595119899(120582 120575 120588 119904)

times [(119899 minus 120572120601

119899)

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

(119899 + 120572120601119899)

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816] 1199032

(32)

by (26)

= (1 minus10038161003816100381610038161198871

1003816100381610038161003816) minus1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(33)

The following covering result follows from the left-handinequality inTheorem 5

Corollary 6 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

120596 |120596| lt21205952minus 1 minus (120595

2(120582 120575 120588 119904) minus 1) 120572

1205952(120582 120575 120588 s) (2 minus 120572)

minus21205952(120582 120575 120588 119904) minus 1 minus (120595

2(120582 120575 120588 119904) + 1) 120572

1205952(120582 120575 120588 119904) (2 minus 120572)

100381610038161003816100381611988711003816100381610038161003816

sub 119891 (119880)

(34)

Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper

References

[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984

[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990

[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006

[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979

[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004

[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007

[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Mathematics and Mathematical Sciences

where

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

(119896 ge 1 120576119896

= 1) (6)

From the definition of 120601119899we know

120601119899=

1 119899 = 120580119896 + 1

0 119899 = 120580119896 + 1(119899 ge 2 120580 119896 ge 1) (7)

The differential operator 119863120590119904

120582120575was introduced by Ali Abu-

baker and Darus [9] We define the differential operator ofthe harmonic function 119891 = ℎ + 119892 given by (5) as

119863120590119904

120582120575119891 (119911) = 119863

120590119904

120582120575ℎ (119911) + (minus1)

119904

119863120590119904

120582120575119892 (119911) (8)

where

119863120590119904

120582120575ℎ (119911) = 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 119886

119899119911119899

119863120590119904

120582120575119892 (119911) =

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 119887

119899119911119899

(9)

and also 120595119899(120582 120575 120590 119904) = 119899

119904

(119862(120575 119899)[1 + 120582(119899 minus 1)])120590 120582 ge 0

119862(120575 119899) = (120575 + 1)119899minus1

(119899 minus 1) for 120575 120590 119904 isin 1198730= 0 1 2

and (119909)119899is the Pochhammer symbol defined by

(119909)119899=

Γ (119909 + 119899)

Γ (119909)

= 1 119899 = 0

119909 (119909 + 1) sdot sdot sdot (119909 + 119899 minus 1) 119899 = 1 2 3

(10)

We note that when 119904 = 0 120590 = 1 and 120582 = 0we obtain the Rus-cheweyh derivative for harmonic functions (see [7]) when120590 = 0weobtain the Salagean operator for harmonic functions(see [10]) and when 120590 = 1 119904 = 0 we obtain the operator forharmonic functions given by Al-Shaqsi and Darus [11]

Let 119872119867120590119904

119896(120582 120575 120572) denote the class of complex-valued

sense-preserving harmonic univalent functions 119891 of theform (5) which satisfy the condition

Im

(120597120597120579)119863120590119904

120582120575119891 (119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

= R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572

(11)

where 119911 = 119903119890119894120579 0 le 119903 lt 1 0 le 120579 lt 120587 0 le 120572 lt 1 and the

functions 119863120590119904120582120575

ℎ119896and 119863

120590119904

120582120575119892119896are of the form

119863120590119904

120582120575ℎ119896(119911) = 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899119886119899119911119899

119863120590119904

120582120575119892119896(119911) =

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899119887119899119911119899

(12)

Further denote by 119872119867120590119904

119896(120582 120575 120572) the subclasses of

119872119867120590119904

119896(120582 120575 120572) such that the functions ℎ and 119892 in 119891 = ℎ + 119892

are of the form

ℎ (119911) = 119911 minus

infin

sum

119899=2

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

119892 (119911) =

infin

sum

119899=1

10038161003816100381610038161198871198991003816100381610038161003816 119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1

(13)

and the functions ℎ119896and 119892

119896in 119891119896= ℎ119896+ 119892119896are of the form

ℎ119896(119911) = 119911 minus

infin

sum

119899=2

120601119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

119892119896(119911) =

infin

sum

119899=1

120601119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899

10038161003816100381610038161198871

1003816100381610038161003816 lt 1

(14)

In this paper we obtain inclusion properties and coefficientconditions for the class 119872119867

120590119904

119896(120582 120575 120572) A representation

theorem and distortion bounds for the class 119872119867120590119904

119896(120582 120575 120572)

are also established

Lemma 1 (see [12]) Let 119891 = ℎ + 119892 isin 119878119867 if

infin

sum

119899=2

1 minus 120572

2 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

infin

sum

119899=1

1 + 120572

2 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816 le 1 (15)

where 0 le 120572 lt 1 then 119891 is harmonic sense-preserving univa-lent in 119880 and 119891 is starlike harmonic of order 120572

2 Main Results

First we give a meaningful conclusion about the class119872119867120590119904

119896(120582 120575 120572)

Theorem 2 Let 119891 isin 119872119867120590119904

119896(120582 120575 120572) where f is given by (1)

then 119891119896defined by (5) is in 119872119867

120590119904

1(120582 120575 120572) = 119872119867

120590119904

(120582 120575 120572)

Proof Let 119891 isin 119872119867120590119904

119896(120582 120575 120572) Then substituting 119903119890

119894120579 by120576]119903119890119894120579 where 120576

119896

= 1 (] = 0 1 119896 minus 1) in (11) respectivelywe have

Im

(120597120597120579)119863120590119904

120582120575119891 (120576

]119903119890119894120579

)

119863120590119904

120582120575119891119896(120576]119903119890119894120579)

ge 120572 (16)

According to the definition of 119891119896and 120576

119896

= 1 we know119891119896(120576

]119903119890119894120579

) = 120576]119891119896(119903119890119894120579

) for any ] = 0 1 119896minus1 and summingup we can get

Im1

119896

119896minus1

sum

]=0

(120597120597120579)119863120590119904

120582120575119891 (120576

]119903119890119894120579

)

120576]119863120590119904

120582120575119891119896(119903119890119894120579)

= Im

(120597120597120579)119863120590119904

120582120575119891119896(119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

ge 120572

(17)

that is 119891119896isin 119872119867

120590119904

(120582 120575 120572)

International Journal of Mathematics and Mathematical Sciences 3

Next a sufficient coefficient condition for harmonic func-tions in 119872119867

120590119904

(120582 120575 120572) is given

Theorem 3 Let 119891 = ℎ + 119892 with ℎ and 119892 given by (1) and119891119896= ℎ119896+ 119892119896with ℎ

119896and 119892

119896given by (5) Let

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904) [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(18)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 Then 119891 is sense-preserving harmonic univalent in119880 and 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

Proof Sinceinfin

sum

119899=1

[119899 minus 120572

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

le

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572120601119899

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1

=

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(19)

by Lemma 1 we conclude that 119891 is sense-preservingharmonic univalent and starlike in 119880 To prove 119891 isin

119872119867120590119904

119896(120582 120575 120572) according to the condition (11) we need to

show that

Im

(120597120597120579)119863120590119904

120582120575119891 (119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

= R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= R119860 (119911)

119861 (119911)ge 120572

119860 (119911) = 119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 119886

119899119911119899

minus (minus1)119904

times

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 119887

119899119911119899

(20)

119861 (119911) = 119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899119886119899119911119899

+ (minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899119887119899119911119899

(21)

where

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1 (22)

Using the fact that R(120596) ge 120572 if and only if |1 minus 120572 + 120596| ge

|1 + 120572 minus 120596| it suffices to show that

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (23)

On the other hand for119860(119911) and 119861(119911) as given in (20) and(21) respectively we have

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

=

10038161003816100381610038161003816100381610038161003816

(1 minus 120572)119863120590119904

120582120575ℎ119896(119911) + 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 minus 120572)119863120590119904

120582120575119892119896(119911) minus 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

minus

10038161003816100381610038161003816100381610038161003816

(1 + 120572)119863120590119904

120582120575ℎ119896(119911) minus 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 + 120572)119863120590119904

120582120575119892119896(119911) + 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(2 minus 120572) 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899] 119886119899119911119899

minus (minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 minus 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

minus 120572119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899] 119886119899119911119899

minus(minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

4 International Journal of Mathematics and Mathematical Sciences

ge (2 minus 120572) |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003817100381710038171003817119887119899

1003817100381710038171003817 |119911|119899

minus 120572 |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899

= (2 minus 120572) |119911| 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1

ge 2 (1 minus 120572) 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 ge 0

(24)

Note that by substituting the value of 120601119899given by (7) in the

previous inequality above then

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

ge 2 (1 minus 120572)

1 minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572 + 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

minus(119899 minus 1) 119896 + 120572 + 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816]

ge 0

(25)

by (18) Thus concludes the proof of the theorem

Next the condition (18) is also necessary for functions in119872119867120590119904

119896(120582 120575 120572) which is clarified

Theorem 4 Let119891 = ℎ+119892with ℎ and 119892 given by (13) and 119891119896=

ℎ119896+119892119896with ℎ

119896and 119892

119896given by (14) Then 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

if and only ifinfin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120588 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(26)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 given by (6)

Proof The if part follows fromTheorem 3 upon noting that ifthe analytic and coanalytic parts of119891 = ℎ+119892 isin 119872119867

120590119904

119896(120582 120575 120572)

are of the form (13) then 119891 isin 119872119867120590119904

119896(120582 120575 120572) For the only if

part we show that 119891 notin 119872119867120590119904

119896(120582 120575 120572) if the condition (26)

does not hold Thus we can write

R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572 (27)

this is equivalent to

0 le R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

minus 120572 (28)

That is

R

((1 minus 120572) 119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119911119899

minus(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119911119899)

times (119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

+(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899)

minus1

ge 0

(29)

The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have

((1 minus 120572) minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119903119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119903119899minus1

)

International Journal of Mathematics and Mathematical Sciences 5

times (1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1

+

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119903119899minus1

)

minus1

ge 0

(30)

If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis

contradicts the required condition for 119891 isin 119872119867120590119904

119896(120582 120575 120572)

and the proof is complete

Now the distortion result is given

Theorem 5 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) minus

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

1003816100381610038161003816119891 (119911)1003816100381610038161003816 le (1 +

100381610038161003816100381611988711003816100381610038161003816) +

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(31)

Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867

120590119904

119896(120582 120575 120572) Taking

the absolute value of 119891(119911) we obtain

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 119903119899

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

times

infin

sum

119899=2

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

2 minus 120572

infin

sum

119899=2

120595119899(120582 120575 120588 119904)

times [(119899 minus 120572120601

119899)

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

(119899 + 120572120601119899)

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816] 1199032

(32)

by (26)

= (1 minus10038161003816100381610038161198871

1003816100381610038161003816) minus1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(33)

The following covering result follows from the left-handinequality inTheorem 5

Corollary 6 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

120596 |120596| lt21205952minus 1 minus (120595

2(120582 120575 120588 119904) minus 1) 120572

1205952(120582 120575 120588 s) (2 minus 120572)

minus21205952(120582 120575 120588 119904) minus 1 minus (120595

2(120582 120575 120588 119904) + 1) 120572

1205952(120582 120575 120588 119904) (2 minus 120572)

100381610038161003816100381611988711003816100381610038161003816

sub 119891 (119880)

(34)

Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper

References

[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984

[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990

[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006

[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979

[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004

[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007

[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 3

Next a sufficient coefficient condition for harmonic func-tions in 119872119867

120590119904

(120582 120575 120572) is given

Theorem 3 Let 119891 = ℎ + 119892 with ℎ and 119892 given by (1) and119891119896= ℎ119896+ 119892119896with ℎ

119896and 119892

119896given by (5) Let

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904) [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(18)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 Then 119891 is sense-preserving harmonic univalent in119880 and 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

Proof Sinceinfin

sum

119899=1

[119899 minus 120572

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

le

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

119899 + 120572120601119899

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816]

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1

=

infin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(19)

by Lemma 1 we conclude that 119891 is sense-preservingharmonic univalent and starlike in 119880 To prove 119891 isin

119872119867120590119904

119896(120582 120575 120572) according to the condition (11) we need to

show that

Im

(120597120597120579)119863120590119904

120582120575119891 (119903119890119894120579

)

119863120590119904

120582120575119891119896(119903119890119894120579)

= R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= R119860 (119911)

119861 (119911)ge 120572

119860 (119911) = 119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 119886

119899119911119899

minus (minus1)119904

times

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 119887

119899119911119899

(20)

119861 (119911) = 119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

= 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899119886119899119911119899

+ (minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899119887119899119911119899

(21)

where

120601119899=

1

119896

119896minus1

sum

]=0120576(119899minus1)]

120576119896

= 1 (22)

Using the fact that R(120596) ge 120572 if and only if |1 minus 120572 + 120596| ge

|1 + 120572 minus 120596| it suffices to show that

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (23)

On the other hand for119860(119911) and 119861(119911) as given in (20) and(21) respectively we have

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

=

10038161003816100381610038161003816100381610038161003816

(1 minus 120572)119863120590119904

120582120575ℎ119896(119911) + 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 minus 120572)119863120590119904

120582120575119892119896(119911) minus 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

minus

10038161003816100381610038161003816100381610038161003816

(1 + 120572)119863120590119904

120582120575ℎ119896(119911) minus 119911(119863

120590119904

120582120575ℎ (119911))

1015840

+(minus1)119904

[(1 + 120572)119863120590119904

120582120575119892119896(119911) + 119911(119863

120590119904

120582120575119892 (119911))

1015840

]

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(2 minus 120572) 119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899] 119886119899119911119899

minus (minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 minus 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus

1003816100381610038161003816100381610038161003816100381610038161003816

minus 120572119911 +

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899] 119886119899119911119899

minus(minus1)119904

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899] 119887119899119911119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

4 International Journal of Mathematics and Mathematical Sciences

ge (2 minus 120572) |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003817100381710038171003817119887119899

1003817100381710038171003817 |119911|119899

minus 120572 |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899

= (2 minus 120572) |119911| 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1

ge 2 (1 minus 120572) 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 ge 0

(24)

Note that by substituting the value of 120601119899given by (7) in the

previous inequality above then

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

ge 2 (1 minus 120572)

1 minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572 + 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

minus(119899 minus 1) 119896 + 120572 + 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816]

ge 0

(25)

by (18) Thus concludes the proof of the theorem

Next the condition (18) is also necessary for functions in119872119867120590119904

119896(120582 120575 120572) which is clarified

Theorem 4 Let119891 = ℎ+119892with ℎ and 119892 given by (13) and 119891119896=

ℎ119896+119892119896with ℎ

119896and 119892

119896given by (14) Then 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

if and only ifinfin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120588 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(26)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 given by (6)

Proof The if part follows fromTheorem 3 upon noting that ifthe analytic and coanalytic parts of119891 = ℎ+119892 isin 119872119867

120590119904

119896(120582 120575 120572)

are of the form (13) then 119891 isin 119872119867120590119904

119896(120582 120575 120572) For the only if

part we show that 119891 notin 119872119867120590119904

119896(120582 120575 120572) if the condition (26)

does not hold Thus we can write

R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572 (27)

this is equivalent to

0 le R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

minus 120572 (28)

That is

R

((1 minus 120572) 119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119911119899

minus(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119911119899)

times (119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

+(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899)

minus1

ge 0

(29)

The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have

((1 minus 120572) minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119903119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119903119899minus1

)

International Journal of Mathematics and Mathematical Sciences 5

times (1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1

+

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119903119899minus1

)

minus1

ge 0

(30)

If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis

contradicts the required condition for 119891 isin 119872119867120590119904

119896(120582 120575 120572)

and the proof is complete

Now the distortion result is given

Theorem 5 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) minus

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

1003816100381610038161003816119891 (119911)1003816100381610038161003816 le (1 +

100381610038161003816100381611988711003816100381610038161003816) +

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(31)

Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867

120590119904

119896(120582 120575 120572) Taking

the absolute value of 119891(119911) we obtain

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 119903119899

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

times

infin

sum

119899=2

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

2 minus 120572

infin

sum

119899=2

120595119899(120582 120575 120588 119904)

times [(119899 minus 120572120601

119899)

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

(119899 + 120572120601119899)

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816] 1199032

(32)

by (26)

= (1 minus10038161003816100381610038161198871

1003816100381610038161003816) minus1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(33)

The following covering result follows from the left-handinequality inTheorem 5

Corollary 6 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

120596 |120596| lt21205952minus 1 minus (120595

2(120582 120575 120588 119904) minus 1) 120572

1205952(120582 120575 120588 s) (2 minus 120572)

minus21205952(120582 120575 120588 119904) minus 1 minus (120595

2(120582 120575 120588 119904) + 1) 120572

1205952(120582 120575 120588 119904) (2 minus 120572)

100381610038161003816100381611988711003816100381610038161003816

sub 119891 (119880)

(34)

Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper

References

[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984

[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990

[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006

[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979

[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004

[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007

[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Mathematics and Mathematical Sciences

ge (2 minus 120572) |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 minus 120572) 120601

119899]1003817100381710038171003817119887119899

1003817100381710038171003817 |119911|119899

minus 120572 |119911| minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [119899 minus (1 + 120572) 120601

119899]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [119899 + (1 + 120572) 120601

119899]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899

= (2 minus 120572) |119911| 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1

ge 2 (1 minus 120572) 1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) [

119899 minus 120572120601119899

1 minus 120572]1003816100381610038161003816119886119899

1003816100381610038161003816

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) [

119899 + 120572120601119899

1 minus 120572]1003816100381610038161003816119887119899

1003816100381610038161003816 ge 0

(24)

Note that by substituting the value of 120601119899given by (7) in the

previous inequality above then

|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|

ge 2 (1 minus 120572)

1 minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904)

times [(119899 minus 1) 119896 minus 120572 + 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

minus(119899 minus 1) 119896 + 120572 + 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816]

ge 0

(25)

by (18) Thus concludes the proof of the theorem

Next the condition (18) is also necessary for functions in119872119867120590119904

119896(120582 120575 120572) which is clarified

Theorem 4 Let119891 = ℎ+119892with ℎ and 119892 given by (13) and 119891119896=

ℎ119896+119892119896with ℎ

119896and 119892

119896given by (14) Then 119891 isin 119872119867

120590119904

119896(120582 120575 120572)

if and only ifinfin

sum

119899=1

120595(119899minus1)119896+1

(120582 120575 120588 119904)

times [(119899 minus 1) 119896 minus 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119886(119899minus1)119896+11003816100381610038161003816

+(119899 minus 1) 119896 + 120572120601

119899+ 1

1 minus 120572

1003816100381610038161003816119887(119899minus1)119896+11003816100381610038161003816]

+

infin

sum

119899=2

119899 =120580119896+1

119899120595119899(120582 120575 120590 119904)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] le 2

(26)

where 1198861= 1 120580 ge 1 and 120582 ge 0 for 120575 120590 119904 isin 119873

0= 0 1 2

and 119896 ge 1 given by (6)

Proof The if part follows fromTheorem 3 upon noting that ifthe analytic and coanalytic parts of119891 = ℎ+119892 isin 119872119867

120590119904

119896(120582 120575 120572)

are of the form (13) then 119891 isin 119872119867120590119904

119896(120582 120575 120572) For the only if

part we show that 119891 notin 119872119867120590119904

119896(120582 120575 120572) if the condition (26)

does not hold Thus we can write

R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

ge 120572 (27)

this is equivalent to

0 le R

119911(119863120590119904

120582120575ℎ (119911))

1015840

minus (minus1)119904

119911(119863120590119904

120582120575119892 (119911))

1015840

119863120590119904

120582120575ℎ119896(119911) + (minus1)

119904

119863120590119904

120582120575119892119896(119911)

minus 120572 (28)

That is

R

((1 minus 120572) 119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119911119899

minus(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119911119899)

times (119911 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119911119899

+(minus1)119904

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119911119899)

minus1

ge 0

(29)

The above-required condition must hold for all values of 119911|119911| = 119903 lt 1 Upon choosing the values of 119911 on the positivereal axis where 0 le 119911 = 119903 lt 1 we must have

((1 minus 120572) minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) (119899 minus 120572120601

119899)1003816100381610038161003816119886119899

1003816100381610038161003816 119903119899minus1

minus

infin

sum

119899=1

120595119899(120582 120575 120590 119904) (119899 + 120572120601

119899)1003816100381610038161003816119887119899

1003816100381610038161003816 119903119899minus1

)

International Journal of Mathematics and Mathematical Sciences 5

times (1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1

+

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119903119899minus1

)

minus1

ge 0

(30)

If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis

contradicts the required condition for 119891 isin 119872119867120590119904

119896(120582 120575 120572)

and the proof is complete

Now the distortion result is given

Theorem 5 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) minus

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

1003816100381610038161003816119891 (119911)1003816100381610038161003816 le (1 +

100381610038161003816100381611988711003816100381610038161003816) +

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(31)

Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867

120590119904

119896(120582 120575 120572) Taking

the absolute value of 119891(119911) we obtain

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 119903119899

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

times

infin

sum

119899=2

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

2 minus 120572

infin

sum

119899=2

120595119899(120582 120575 120588 119904)

times [(119899 minus 120572120601

119899)

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

(119899 + 120572120601119899)

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816] 1199032

(32)

by (26)

= (1 minus10038161003816100381610038161198871

1003816100381610038161003816) minus1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(33)

The following covering result follows from the left-handinequality inTheorem 5

Corollary 6 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

120596 |120596| lt21205952minus 1 minus (120595

2(120582 120575 120588 119904) minus 1) 120572

1205952(120582 120575 120588 s) (2 minus 120572)

minus21205952(120582 120575 120588 119904) minus 1 minus (120595

2(120582 120575 120588 119904) + 1) 120572

1205952(120582 120575 120588 119904) (2 minus 120572)

100381610038161003816100381611988711003816100381610038161003816

sub 119891 (119880)

(34)

Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper

References

[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984

[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990

[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006

[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979

[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004

[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007

[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 5

times (1 minus

infin

sum

119899=2

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1

+

infin

sum

119899=1

120595119899(120582 120575 120590 119904) 120601

119899

10038161003816100381610038161198871198991003816100381610038161003816 119903119899minus1

)

minus1

ge 0

(30)

If the condition (26) does not hold then the numerator in(30) is negative for 119903 sufficiently close to 1 Hence there exists a1199110= 1199030in (0 1) for which the quotient in (30) is negativeThis

contradicts the required condition for 119891 isin 119872119867120590119904

119896(120582 120575 120572)

and the proof is complete

Now the distortion result is given

Theorem 5 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) minus

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

1003816100381610038161003816119891 (119911)1003816100381610038161003816 le (1 +

100381610038161003816100381611988711003816100381610038161003816) +

1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(31)

Proof We will only prove the left-hand inequality of theabove theorem The arguments for the right-hand inequalityare similar and so we omit it Let 119891 isin 119872119867

120590119904

119896(120582 120575 120572) Taking

the absolute value of 119891(119911) we obtain

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge (1 minus

100381610038161003816100381611988711003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 119903119899

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus

infin

sum

119899=2

[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

times

infin

sum

119899=2

1205952(120582 120575 120588 119904) (2 minus 120572120601

2)

1 minus 120572[1003816100381610038161003816119886119899

1003816100381610038161003816 +1003816100381610038161003816119887119899

1003816100381610038161003816] 1199032

ge (1 minus10038161003816100381610038161198871

1003816100381610038161003816) 119903 minus1 minus 120572

2 minus 120572

infin

sum

119899=2

120595119899(120582 120575 120588 119904)

times [(119899 minus 120572120601

119899)

1 minus 120572

10038161003816100381610038161198861198991003816100381610038161003816 +

(119899 + 120572120601119899)

1 minus 120572

10038161003816100381610038161198871198991003816100381610038161003816] 1199032

(32)

by (26)

= (1 minus10038161003816100381610038161198871

1003816100381610038161003816) minus1

1205952(120582 120575 120590 119904)

[1 minus 120572

2 minus 120572minus

1 + 120572

2 minus 120572

100381610038161003816100381611988711003816100381610038161003816] 1199032

(33)

The following covering result follows from the left-handinequality inTheorem 5

Corollary 6 If 119891 isin 119872119867120590119904

119896(120582 120575 120572) then

120596 |120596| lt21205952minus 1 minus (120595

2(120582 120575 120588 119904) minus 1) 120572

1205952(120582 120575 120588 s) (2 minus 120572)

minus21205952(120582 120575 120588 119904) minus 1 minus (120595

2(120582 120575 120588 119904) + 1) 120572

1205952(120582 120575 120588 119904) (2 minus 120572)

100381610038161003816100381611988711003816100381610038161003816

sub 119891 (119880)

(34)

Note that other work related to Sakaguchi and classes offunctions with respect to symmetric points can be found in[13ndash16]

Conflict of Interests

The authors declare that they have no competing interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments given to improve the paper

References

[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A IMathematica vol9 pp 3ndash25 1984

[2] T Sheil-Small ldquoConstants for planar harmonic mappingsrdquoJournal of the London Mathematical Society vol 42 no 2 pp237ndash248 1990

[3] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[4] T N Shanmugam C Ramachandran and V RavichandranldquoFekete-Szego problem for subclasses of starlike functions withrespect to symmetric pointsrdquo Bulletin of the Korean Mathemat-ical Society vol 43 no 3 pp 589ndash598 2006

[5] R Chand and P Singh ldquoOn certain schlicht mappingsrdquo IndianJournal of Pure and AppliedMathematics vol 10 no 9 pp 1167ndash1174 1979

[6] R N das and P Singh ldquoOn subclasses of schlicht mappingrdquoIndian Journal of Pure and Applied Mathematics vol 8 no 8pp 864ndash872 1977

[7] O P Ahuja and J M Jahangiri ldquoSakaguchi-type harmonicunivalent functionsrdquo Scientiae Mathematicae Japonicae vol 59no 1 pp 239ndash244 2004

[8] K Al Shaqsi and M Darus ldquoOn subclass of harmonic starlikefunctions with respect to 119870-symmetric pointsrdquo InternationalMathematical Forum vol 2 no 57-60 pp 2799ndash2805 2007

[9] AAAli Abubaker andMDarus ldquoOn starlike and convex func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematics and Mathematical Sciences vol 2011 Article ID834064 9 pages 2011

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Mathematics and Mathematical Sciences

[10] J M Jahangiri G Murugusundaramoorthy and K VijayaldquoSalagean-type harmonic univalent functionsrdquo Southwest Jour-nal of Pure and Applied Mathematics no 2 pp 77ndash82 2002

[11] K Al-Shaqsi and M Darus ldquoOn univalent functions withrespect to 119896-symmetric points defined by a generalized Rus-cheweyh derivatives operatorrdquo Journal of Analysis and Applica-tions vol 7 no 1 pp 53ndash61 2009

[12] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999

[13] A Abubaker andMDarus ldquoA note on subclass of analytic func-tions with respect to 119896-symmetric pointsrdquo International Journalof Mathematical Analysis vol 6 no 12 pp 573ndash589 2012

[14] K Al-Shaqsi and M Darus ldquoApplication of Holder inequalityin generalised convolutions for functions with respect to 119896-symmetric pointsrdquoAppliedMathematical Sciences vol 3 no 33-36 pp 1787ndash1797 2009

[15] M Darus and R W Ibrahim ldquoAnalytic functions with respectto 119873-symmetric conjugate and symmetric conjugate pointsinvolving generalized differential operatorrdquo Proceedings of thePakistan Academy of Sciences vol 46 no 2 pp 75ndash83 2009

[16] B A Frasin andM Darus ldquoSubordination results on subclassesconcerning Sakaguchi functionsrdquo Journal of Inequalities andApplications vol 2009 Article ID 574014 7 pages 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of