Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 847287 6 pageshttpdxdoiorg1011552013847287

Research ArticleCertain Properties of a Class of Close-to-Convex FunctionsRelated to Conic Domains

Wasim Ul-Haq and Shahid Mahmood

Department of Mathematics Abdul Wali Khan University Mardan Mardan 23200 Pakistan

Correspondence should be addressed to Wasim Ul-Haq wasim474hotmailcom

Received 30 September 2012 Revised 3 March 2013 Accepted 17 March 2013

Academic Editor Nikolaos Papageorgiou

Copyright copy 2013 W Ul-Haq and S MahmoodThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We aim to define a new class of close-to-convex functions which is related to conic domains Many interesting properties such assufficiency criteria inclusion results and integral preserving properties are investigated here Some interesting consequences of ourresults are also observed

1 Introduction

LetA be the class of functions 119891

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 (1)

which are analytic in the open unit disc 119864 = 119911 isin C |119911| lt1 Let 119891 and 119892 be analytic in 119864 and we say that 119891 is sub-ordinate to 119892 written as 119891(119911) ≺ 119892(119911) if there exists a Schwarzfunction119908 which is analytic in 119864with119908(0) = 0 and |119908(119911)| lt1 (119911 isin 119864) such that 119891(119911) = 119892(119908(119911)) In particular when119892 is univalent then the above subordination is equivalent to119891(0) = 119892(0) and 119891(119864) sube 119892(119864) see [1]

Kanas and Wisniowska [2 3] introduced and studied theclasses of 119896-uniformly convex functions denoted by 119896-UCVand the corresponding class 119896-ST related by the Alexander-type relation Later the class 119896-uniformly close-to-convexfunctions denoted by 119896-UK defined as

119896-UK = 119891 isin A

Re(1199111198911015840 (119911)

119892 (119911)) gt 119896

100381610038161003816100381610038161003816100381610038161003816

1199111198911015840 (119911)

119892 (119911)minus 1

100381610038161003816100381610038161003816100381610038161003816

119892 (119911) isin 119896-ST 119911 isin 119864

(2)

was considered by Acu [4] for study details on these classeswe refer to [5ndash7] All these abovementioned classes were gen-eralized to the classes SD(119896 120575) CD(119896 120575) and KD(119896 120573 120575)by Shams et al [8] and Srivastava et al [9] respectively Theclasses SD(119896 120575) andCD(119896 120575) are defined as

SD (119896 120575) = 119891 isin A

Re(1199111198911015840 (119911)

119891 (119911)) gt 119896

100381610038161003816100381610038161003816100381610038161003816

1199111198911015840 (119911)

119891 (119911)minus 1

100381610038161003816100381610038161003816100381610038161003816

+ 120575 119911 isin 119864

CD (119896 120575) = 119891 isin A

Re(1 +11991111989110158401015840 (119911)

1198911015840 (119911)) gt 119896

100381610038161003816100381610038161003816100381610038161003816

11991111989110158401015840 (119911)

1198911015840 (119911)

100381610038161003816100381610038161003816100381610038161003816

+ 120575 119911 isin 119864

(3)

where 119896 ge 0 0 le 120573 120575 lt 1 The class KD(119896 120573 120575) knownas 119896-uniformly close-to-convex functions of order 120573 type 120575

2 Abstract and Applied Analysis

is the class of all those functions 119891 isin A which satisfies thefollowing condition

Re(1198911015840 (119911)

1198921015840 (119911)) ge 119896

100381610038161003816100381610038161003816100381610038161003816

1198911015840 (119911)

1198921015840 (119911)minus 1

100381610038161003816100381610038161003816100381610038161003816+ 120573

(119896 ge 0 0 le 120573 120575 lt 1 119911 isin 119864)

(4)

for some 119892 isin CD(119896 120575)Motivated by the work of Noor et al [10ndash13] we define

the following

Definition 1 Let 119891 isin A Then 119891 is in the class TD(119896 120572 120573120574 120575) if and only if for 120572 ge 0 0 le 120573 120574 120575 lt 1

Re (119867 (120572 120573 120574 119891 119892)) ge 1198961003816100381610038161003816119867 (120572 120573 120574 119891 119892) minus 1

1003816100381610038161003816

(119896 ge 0 119911 isin 119864) (5)

for some 119892 isin CD(119896 120575) where

119867(120572 120573 120574 119891 119892) =1 minus 120572

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+120572

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(6)

Special Cases

(i) (119896 0 120573 120574 120575) = KD(119896 120573 120575) see [9]

(ii) TD(1 0 120573 120574 0) = UK(120573) and TD(1 1 120573 120574 0) =UQ(120574) the classes of uniformly close-to-convex andquasiconvex functions introduced and investigated in[14]

(iii) TD(0 120572 0 0 0) = Q120572 the class of alpha quasiconvex

functions introduced and studied in [11]

(iv) TD(0 0 120573 120574 120575) = K(120573 120575) the class of close-to-convex functions of order 120573 type 120575 [15]

(v) TD(0 1 120573 120574 120575) = 119862lowast(120574 120575) the class of quasiconvexfunctions of order 120574 type 120575 [16]

The conditions 119896 ge 0 120572 ge 0 0 le 120573 120574 120575 lt 1 on the para-meters are assumed throughout the entire paper unless other-wise mentionedGeometric Interpretation A function 119891 isin A is in the classTD(119896 120572 120573 120574 120575) if and only if the functional 119867(120572 120573 120574 120575 119891119892) takes all the values in the conic domain Ω

119896defined as

follows

Ω119896= 119906 + 120580V 119906 gt 119896radic(119906 minus 1)2 + V2 (7)

Extremal functions for these conic regions are denoted by119901119896(119911) which are analytic in 119864 and map 119864 onto Ω

119896such that

119901119896(0) = 1 and 1199011015840

119896(0) gt 1 These functions are given as

119901119896(119911) =

1 + 119911

1 minus 119911 119896 = 0

1 +2

1205872(log 1 + radic119911

1 minus radic119911)2

119896 = 1

1 +2

1 minus 1198962sinh2 [( 2

120587arccos 119896) arctan ℎradic119911]

0 lt 119896 lt 1

1+1

1198962minus1sin( 120587

2119877 (119905)int119906(119911)radic119905

0

1

radic1minus1199092radic1minus(119905119909)2119889119909)

+1

1198962 minus 1 119896 gt 1

(8)

where 119906(119911) = (119911 minus radic119905)(1 minus radic119905119911) 119905 isin (0 1) 119911 isin 119864 and119911 is chosen such that 119896 = cosh(1205871198771015840(119905)4119877(119905)) where 119877(119905) isLegendrersquos complete elliptic integral of the first kind and1198771015840(119905)is complementary integral of 119877(119905) see [2 3]

2 Preliminaries Result

We require the following results which are essential in ourinvestigations

Lemma 2 (see [17 page 70]) Let ℎ be convex function in 119864and 119902 119864 997891rarr 119862 with Re 119902(119911) gt 0 119911 isin 119864 If 119901 is analytic in 119864with 119901(0) = ℎ(0) then

119901 (119911) + 119902 (119911) 1199111199011015840

(119911) ≺ ℎ (119911) implies 119901 (119911) ≺ ℎ (119911) (9)

Lemma 3 (see [17 page 195]) Let ℎ be convex function in 119864with ℎ(0) = 0 and 119860 ge 0 Suppose that 119895 ge 4ℎ1015840(0) and that119861(119911) 119862(119911) and 119863(119911) are analytic in 119864 and satisfy

Re119861 (119911) ge 119860 + |119862 (119911) minus 1| minus Re (119862 (119911) minus 1) + 119895119863 (119911) (10)

for 119911 isin 119864 If119901 is analytic in119864with119901(119911) = 1198861119911+11988621199112+11988631199113+sdot sdot sdot

and the following subordination relation holds

119860119911211990110158401015840 (119911) + 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ ℎ (119911)

(11)

then

119901 (119911) ≺ ℎ (119911) (12)

Lemma 4 (see [12]) If 119891(119911) ≺ ℎ(119911) and 119892(119911) ≺ ℎ(119911) then for120572 isin [0 1]

(1 minus 120572) 119891 (119911) + 120572119892 (119911) ≺ ℎ (119911) (13)

3 Main Results

First we prove the following sufficiency criteria for the func-tions in the classTD(119896 120572 120573 120574 120575)

Abstract and Applied Analysis 3

Theorem 5 A function 119891 isin A is said to be in the classTD(119896120572 120573 120574 120575) if

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) lt (1 minus 120573) (1 minus 120574) (14)

where

Φ119899(119896 120572 120573 120574 120575)

= (119896 + 1) [(1 minus 120572) (1 minus 120574) 119899 +120572 (1 minus 120573) 1198992]1003816100381610038161003816119886119899

1003816100381610038161003816

+ [(119896 + 2 minus 120573) (1 minus 120574) +120572 (119896 + 1) (120574 minus 120573) 1198991003816100381610038161003816119887119899

1003816100381610038161003816]

(15)

Proof Let us assume that relation (6) holds Now it is suffi-cient to show that

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892)minus 1

1003816100381610038161003816minus Re [119867 (120572 120573 120574 120575 119891 119892)minus 1] lt 1(16)

First we consider

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+120572

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

minus 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573

1198911015840 (119911)

1198921015840 (119911)+

120572

1 minus 120574

(1199111198911015840 (119911))1015840

1198921015840 (119911)

minus(1 minus 120572) 120573

1 minus 120573minus

120572120574

1 minus 120574minus 1

10038161003816100381610038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) 1198911015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

+120572 (1 minus 120573) (1199111198911015840 (119911))

1015840

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] 1198921015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

100381610038161003816100381610038161003816100381610038161003816

(17)

Using (1) and the series 119892(119911) = 119911 + suminfin

119899=2119887119899119911119899 in (17) we have

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (1 + suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (1 + sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (1 + sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

(18)

Now

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816 minus Re [119867 (120572 120573 120574 120575 119891 119892) minus 1]

le (119896 + 1)1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816

le (119896 + 1) [(1 minus 120572) (1 minus 120574) (sum

infin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)]

(19)

4 Abstract and Applied Analysis

The last inequality is bounded above by 1 if

(119896 + 1) [(1 minus 120572) (1 minus 120574)(infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816) + 120572 (1 minus 120573)(infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816)

+ [(1 minus 120574) + 120572 (120574 minus 120573)] (infin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)]

le (1 minus 120573) (1 minus 120574)(1 minusinfin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)

(20)

Hence

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) le (1 minus 120573) (1 minus 120574) (21)

where Φ119899(119896 120572 120573 120574 120575) is given by (15) This completes the

proof

When we take 120572 = 0 119896 = 1 and 119892(119911) = 119911 in the abovetheorem we obtain the following sufficient condition for thefunctions to be in the classUK(120573) which is proved in [14]

Corollary 6 (see [14]) A function 119891 isin 119860 is said to be in theclassUK(120573) if

infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120573)

2 (22)

Corollary 7 (see [14]) A function 119891 isin 119860 is said to be in theclassUQ(120574) if

infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120574)

2 (23)

The above corollary is obtained when we take 120572 = 1 119896 =1 and 119892(119911) = 119911 in Theorem 5

Theorem 8 Let 119891 isin TD(119896 120572 120573 120574 120575) and 120572 ge 4(120573(1+3120573))Then 119891 isin KD(119896 0 120575)

Proof Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) (24)

where 119901(119911) is analytic and 119901(0) = 1 Now differentiating (24)we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 1199111199011015840 (119911) + 119901 (119911) 120595 (119911) (25)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (24) and (25) in relation(6) we obtain

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[119901 (119911) minus 120573] +

120572

1 minus 120574[1199111199011015840 (119911) + 119901 (119911) 120595 (119911) minus 120574]

=120572

1 minus 1205741199111199011015840 (119911)+[

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] 119901 (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911)

(26)

where

119861 (119911)=120572

1minus120574 119862 (119911)=

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1minus120573) (1minus120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(27)

Now since 119891 isin TD(119896 120572 120573 120574 120575) we have

119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ 119901119896(119911) (28)

Replacing 119901(119911) by 119901lowast(119911) = 119901(119911) minus 1 and 119901

119896(119911) by 119901lowast

119896(119911) =

119901119896(119911) minus 1 the above subordination is equivalent to

119861 (119911) 1199111199011015840

lowast(119911) + 119862 (119911) 119901

lowast(119911) + 119863

lowast(119911) ≺ 119901lowast

119896(119911) (29)

where119863lowast(119911) = 119862(119911) + 119863(119911) minus 1 Using Lemma 3 with 119860 = 0

we obtain

119901lowast(119911) ≺ 119901lowast

119896(119911) (30)

This implies that

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) ≺ 119901

119896(119911) (31)

Hence 119891 isin KD(119896 0 120575) This completes the proof

Corollary 9 Let 119891 isin TD(119896 120572 0 0 0) = Q120572 Then 119891 isin

KD(0 0 0) = K That is Q120572sub K 120572 ge 0

The above result is well-known inclusion proved in [11]For 119891 isin 119860 consider the following integral operator de-

fined by

119865 (119911) = 119868119898[119891] =

119898 + 1

119911119898int119911

0

119905119898minus1119891 (119905) 119889119905 119898 = 1 2 3

(32)

This operatorwas given byBernardi [18] in 1969 In particularthe operator 119868

1was considered by Libera [19] Now let us

prove the following

Abstract and Applied Analysis 5

Theorem 10 Let 119891 isin TD(119896 120572 120573 120574 120575) Then 119868119898[119891] isin

KD(119896 0 120575)

Proof Let the function 119892 be such that (6) is satisfied It caneasily be seen that according to [4] the function119866 = 119868

119898[119891] isin

CD(119896 120575) and from (32) we deduce

(1 + 119898)1198911015840

(119911) = (1 + 119898) 1198651015840

(119911) + 11991111986510158401015840 (119911) (33)

(1 + 119898) 1198921015840

(119911) = (1 + 119898)1198661015840

(119911) + 11991111986610158401015840 (119911) (34)

If we let 119901(119911) = 1198651015840(119911)1198661015840(119911) and 119902(119911) = 1(119898 + 1 +(11991111986610158401015840(119911)1198661015840)) then simple computations yield us

1198911015840 (119911)

1198921015840 (119911)=

(1 + 119898) 1198651015840 (119911) + 11991111986510158401015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

=119911119901 (119911) 1198921015840 (119911) + 119911119901 (119911) 11989210158401015840 (119911) + 1199111198921015840 (119911) 1199011015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

= 119901 (119911) + 119902 (119911) 1199111199011015840

(119911)

(35)

Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) + 1199111199011015840 (119911) 119902 (119911) = ℎ (119911) (36)

where ℎ(119911) is analytic and 119901(0) = 1 From (36) we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 120595 (119911) ℎ (119911) + 119911ℎ1015840 (119911) (37)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (36) and (37) in (6) wehave

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[ℎ (119911) minus 120573] +

120572

1 minus 120574[119911ℎ1015840 (119911) + ℎ (119911) 120595 (119911) minus 120574]

=120572

1 minus 120574119911ℎ1015840 (119911)

+ [(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] ℎ (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 119911ℎ1015840

(119911) + 119862 (119911) ℎ (119911) + 119863 (119911)

(38)

where

119861 (119911) =120572

1 minus 120574 119862 (119911) =

(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(39)

Now proceeding in the similar manner as in the proof ofTheorem 5 and using Lemma 3 with 119860 = 0 we obtain

1198911015840 (119911)

1198921015840 (119911)= ℎ (119911) ≺ 119901

119896(119911) (40)

From (36) it implies that

119901 (119911) + 1199111199011015840 (119911) 119902 (119911) ≺ 119901119896(119911) (41)

By employing Lemma 2 we immediately obtain the desiredresult

Theorem 11 For 120572 gt 1205721ge 0

TD (119896 120572 120573 120574 120575) sube TD (119896 1205721 120573 120574 120575) (42)

Proof Let 119891 isin TD(119896 120572 120573 120574 120575) Then consider

119867(1205721 120573 120574 120575 119891 119892) =

1 minus 1205721

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(43)

After some simple computations we have

119867(1205721 120573 120574 120575 119891 119892)

=1

1 minus 120573(1 minus

1205721

120572)[

1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

120572[

[

1 minus 120572

1 minus 120573(1198911015840 (119911)

1198921015840 (119911)minus 120573)+

120572

1 minus 120574((1199111198911015840 (119911))

1015840

1198921015840 (119911)minus 120574)]

]

=(1 minus1205721

120572)119867 (0 120573 120574 120575 119891 119892)+

1205721

120572119867 (120572 120573 120574 120575 119891 119892)

(44)

Now since119891 isin TD(119896 120572 120573 120574 120575) we have119867(120572 120573 120574 120575 119891 119892) ≺119901119896(119911) Also Theorem 8 gives us that 119867(0 120573 120574 120575 119891 119892) ≺

119901119896(119911) The use of Lemma 4 leads us to the required relation

that is 119867(1205721 120573 120574 120575 119891 119892) ≺ 119901

119896(119911) This completes the

proof

Acknowledgment

The authors would like to thank the reviewer of this paperfor hisher valuable comments on the earlier version of thispaper They would also like to acknowledge Prof Dr EhsanAli VCAWKUM for the financial support on the publicationof this article

References

[1] AW GoodmanUnivalent Functions Vol I Polygonal Publish-ing House Washington DC USA 1983

[2] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

6 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

is the class of all those functions 119891 isin A which satisfies thefollowing condition

Re(1198911015840 (119911)

1198921015840 (119911)) ge 119896

100381610038161003816100381610038161003816100381610038161003816

1198911015840 (119911)

1198921015840 (119911)minus 1

100381610038161003816100381610038161003816100381610038161003816+ 120573

(119896 ge 0 0 le 120573 120575 lt 1 119911 isin 119864)

(4)

for some 119892 isin CD(119896 120575)Motivated by the work of Noor et al [10ndash13] we define

the following

Definition 1 Let 119891 isin A Then 119891 is in the class TD(119896 120572 120573120574 120575) if and only if for 120572 ge 0 0 le 120573 120574 120575 lt 1

Re (119867 (120572 120573 120574 119891 119892)) ge 1198961003816100381610038161003816119867 (120572 120573 120574 119891 119892) minus 1

1003816100381610038161003816

(119896 ge 0 119911 isin 119864) (5)

for some 119892 isin CD(119896 120575) where

119867(120572 120573 120574 119891 119892) =1 minus 120572

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+120572

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(6)

Special Cases

(i) (119896 0 120573 120574 120575) = KD(119896 120573 120575) see [9]

(ii) TD(1 0 120573 120574 0) = UK(120573) and TD(1 1 120573 120574 0) =UQ(120574) the classes of uniformly close-to-convex andquasiconvex functions introduced and investigated in[14]

(iii) TD(0 120572 0 0 0) = Q120572 the class of alpha quasiconvex

functions introduced and studied in [11]

(iv) TD(0 0 120573 120574 120575) = K(120573 120575) the class of close-to-convex functions of order 120573 type 120575 [15]

(v) TD(0 1 120573 120574 120575) = 119862lowast(120574 120575) the class of quasiconvexfunctions of order 120574 type 120575 [16]

The conditions 119896 ge 0 120572 ge 0 0 le 120573 120574 120575 lt 1 on the para-meters are assumed throughout the entire paper unless other-wise mentionedGeometric Interpretation A function 119891 isin A is in the classTD(119896 120572 120573 120574 120575) if and only if the functional 119867(120572 120573 120574 120575 119891119892) takes all the values in the conic domain Ω

119896defined as

follows

Ω119896= 119906 + 120580V 119906 gt 119896radic(119906 minus 1)2 + V2 (7)

Extremal functions for these conic regions are denoted by119901119896(119911) which are analytic in 119864 and map 119864 onto Ω

119896such that

119901119896(0) = 1 and 1199011015840

119896(0) gt 1 These functions are given as

119901119896(119911) =

1 + 119911

1 minus 119911 119896 = 0

1 +2

1205872(log 1 + radic119911

1 minus radic119911)2

119896 = 1

1 +2

1 minus 1198962sinh2 [( 2

120587arccos 119896) arctan ℎradic119911]

0 lt 119896 lt 1

1+1

1198962minus1sin( 120587

2119877 (119905)int119906(119911)radic119905

0

1

radic1minus1199092radic1minus(119905119909)2119889119909)

+1

1198962 minus 1 119896 gt 1

(8)

where 119906(119911) = (119911 minus radic119905)(1 minus radic119905119911) 119905 isin (0 1) 119911 isin 119864 and119911 is chosen such that 119896 = cosh(1205871198771015840(119905)4119877(119905)) where 119877(119905) isLegendrersquos complete elliptic integral of the first kind and1198771015840(119905)is complementary integral of 119877(119905) see [2 3]

2 Preliminaries Result

We require the following results which are essential in ourinvestigations

Lemma 2 (see [17 page 70]) Let ℎ be convex function in 119864and 119902 119864 997891rarr 119862 with Re 119902(119911) gt 0 119911 isin 119864 If 119901 is analytic in 119864with 119901(0) = ℎ(0) then

119901 (119911) + 119902 (119911) 1199111199011015840

(119911) ≺ ℎ (119911) implies 119901 (119911) ≺ ℎ (119911) (9)

Lemma 3 (see [17 page 195]) Let ℎ be convex function in 119864with ℎ(0) = 0 and 119860 ge 0 Suppose that 119895 ge 4ℎ1015840(0) and that119861(119911) 119862(119911) and 119863(119911) are analytic in 119864 and satisfy

Re119861 (119911) ge 119860 + |119862 (119911) minus 1| minus Re (119862 (119911) minus 1) + 119895119863 (119911) (10)

for 119911 isin 119864 If119901 is analytic in119864with119901(119911) = 1198861119911+11988621199112+11988631199113+sdot sdot sdot

and the following subordination relation holds

119860119911211990110158401015840 (119911) + 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ ℎ (119911)

(11)

then

119901 (119911) ≺ ℎ (119911) (12)

Lemma 4 (see [12]) If 119891(119911) ≺ ℎ(119911) and 119892(119911) ≺ ℎ(119911) then for120572 isin [0 1]

(1 minus 120572) 119891 (119911) + 120572119892 (119911) ≺ ℎ (119911) (13)

3 Main Results

First we prove the following sufficiency criteria for the func-tions in the classTD(119896 120572 120573 120574 120575)

Abstract and Applied Analysis 3

Theorem 5 A function 119891 isin A is said to be in the classTD(119896120572 120573 120574 120575) if

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) lt (1 minus 120573) (1 minus 120574) (14)

where

Φ119899(119896 120572 120573 120574 120575)

= (119896 + 1) [(1 minus 120572) (1 minus 120574) 119899 +120572 (1 minus 120573) 1198992]1003816100381610038161003816119886119899

1003816100381610038161003816

+ [(119896 + 2 minus 120573) (1 minus 120574) +120572 (119896 + 1) (120574 minus 120573) 1198991003816100381610038161003816119887119899

1003816100381610038161003816]

(15)

Proof Let us assume that relation (6) holds Now it is suffi-cient to show that

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892)minus 1

1003816100381610038161003816minus Re [119867 (120572 120573 120574 120575 119891 119892)minus 1] lt 1(16)

First we consider

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+120572

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

minus 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573

1198911015840 (119911)

1198921015840 (119911)+

120572

1 minus 120574

(1199111198911015840 (119911))1015840

1198921015840 (119911)

minus(1 minus 120572) 120573

1 minus 120573minus

120572120574

1 minus 120574minus 1

10038161003816100381610038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) 1198911015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

+120572 (1 minus 120573) (1199111198911015840 (119911))

1015840

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] 1198921015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

100381610038161003816100381610038161003816100381610038161003816

(17)

Using (1) and the series 119892(119911) = 119911 + suminfin

119899=2119887119899119911119899 in (17) we have

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (1 + suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (1 + sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (1 + sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

(18)

Now

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816 minus Re [119867 (120572 120573 120574 120575 119891 119892) minus 1]

le (119896 + 1)1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816

le (119896 + 1) [(1 minus 120572) (1 minus 120574) (sum

infin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)]

(19)

4 Abstract and Applied Analysis

The last inequality is bounded above by 1 if

(119896 + 1) [(1 minus 120572) (1 minus 120574)(infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816) + 120572 (1 minus 120573)(infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816)

+ [(1 minus 120574) + 120572 (120574 minus 120573)] (infin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)]

le (1 minus 120573) (1 minus 120574)(1 minusinfin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)

(20)

Hence

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) le (1 minus 120573) (1 minus 120574) (21)

where Φ119899(119896 120572 120573 120574 120575) is given by (15) This completes the

proof

When we take 120572 = 0 119896 = 1 and 119892(119911) = 119911 in the abovetheorem we obtain the following sufficient condition for thefunctions to be in the classUK(120573) which is proved in [14]

Corollary 6 (see [14]) A function 119891 isin 119860 is said to be in theclassUK(120573) if

infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120573)

2 (22)

Corollary 7 (see [14]) A function 119891 isin 119860 is said to be in theclassUQ(120574) if

infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120574)

2 (23)

The above corollary is obtained when we take 120572 = 1 119896 =1 and 119892(119911) = 119911 in Theorem 5

Theorem 8 Let 119891 isin TD(119896 120572 120573 120574 120575) and 120572 ge 4(120573(1+3120573))Then 119891 isin KD(119896 0 120575)

Proof Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) (24)

where 119901(119911) is analytic and 119901(0) = 1 Now differentiating (24)we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 1199111199011015840 (119911) + 119901 (119911) 120595 (119911) (25)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (24) and (25) in relation(6) we obtain

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[119901 (119911) minus 120573] +

120572

1 minus 120574[1199111199011015840 (119911) + 119901 (119911) 120595 (119911) minus 120574]

=120572

1 minus 1205741199111199011015840 (119911)+[

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] 119901 (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911)

(26)

where

119861 (119911)=120572

1minus120574 119862 (119911)=

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1minus120573) (1minus120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(27)

Now since 119891 isin TD(119896 120572 120573 120574 120575) we have

119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ 119901119896(119911) (28)

Replacing 119901(119911) by 119901lowast(119911) = 119901(119911) minus 1 and 119901

119896(119911) by 119901lowast

119896(119911) =

119901119896(119911) minus 1 the above subordination is equivalent to

119861 (119911) 1199111199011015840

lowast(119911) + 119862 (119911) 119901

lowast(119911) + 119863

lowast(119911) ≺ 119901lowast

119896(119911) (29)

where119863lowast(119911) = 119862(119911) + 119863(119911) minus 1 Using Lemma 3 with 119860 = 0

we obtain

119901lowast(119911) ≺ 119901lowast

119896(119911) (30)

This implies that

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) ≺ 119901

119896(119911) (31)

Hence 119891 isin KD(119896 0 120575) This completes the proof

Corollary 9 Let 119891 isin TD(119896 120572 0 0 0) = Q120572 Then 119891 isin

KD(0 0 0) = K That is Q120572sub K 120572 ge 0

The above result is well-known inclusion proved in [11]For 119891 isin 119860 consider the following integral operator de-

fined by

119865 (119911) = 119868119898[119891] =

119898 + 1

119911119898int119911

0

119905119898minus1119891 (119905) 119889119905 119898 = 1 2 3

(32)

This operatorwas given byBernardi [18] in 1969 In particularthe operator 119868

1was considered by Libera [19] Now let us

prove the following

Abstract and Applied Analysis 5

Theorem 10 Let 119891 isin TD(119896 120572 120573 120574 120575) Then 119868119898[119891] isin

KD(119896 0 120575)

Proof Let the function 119892 be such that (6) is satisfied It caneasily be seen that according to [4] the function119866 = 119868

119898[119891] isin

CD(119896 120575) and from (32) we deduce

(1 + 119898)1198911015840

(119911) = (1 + 119898) 1198651015840

(119911) + 11991111986510158401015840 (119911) (33)

(1 + 119898) 1198921015840

(119911) = (1 + 119898)1198661015840

(119911) + 11991111986610158401015840 (119911) (34)

If we let 119901(119911) = 1198651015840(119911)1198661015840(119911) and 119902(119911) = 1(119898 + 1 +(11991111986610158401015840(119911)1198661015840)) then simple computations yield us

1198911015840 (119911)

1198921015840 (119911)=

(1 + 119898) 1198651015840 (119911) + 11991111986510158401015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

=119911119901 (119911) 1198921015840 (119911) + 119911119901 (119911) 11989210158401015840 (119911) + 1199111198921015840 (119911) 1199011015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

= 119901 (119911) + 119902 (119911) 1199111199011015840

(119911)

(35)

Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) + 1199111199011015840 (119911) 119902 (119911) = ℎ (119911) (36)

where ℎ(119911) is analytic and 119901(0) = 1 From (36) we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 120595 (119911) ℎ (119911) + 119911ℎ1015840 (119911) (37)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (36) and (37) in (6) wehave

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[ℎ (119911) minus 120573] +

120572

1 minus 120574[119911ℎ1015840 (119911) + ℎ (119911) 120595 (119911) minus 120574]

=120572

1 minus 120574119911ℎ1015840 (119911)

+ [(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] ℎ (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 119911ℎ1015840

(119911) + 119862 (119911) ℎ (119911) + 119863 (119911)

(38)

where

119861 (119911) =120572

1 minus 120574 119862 (119911) =

(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(39)

Now proceeding in the similar manner as in the proof ofTheorem 5 and using Lemma 3 with 119860 = 0 we obtain

1198911015840 (119911)

1198921015840 (119911)= ℎ (119911) ≺ 119901

119896(119911) (40)

From (36) it implies that

119901 (119911) + 1199111199011015840 (119911) 119902 (119911) ≺ 119901119896(119911) (41)

By employing Lemma 2 we immediately obtain the desiredresult

Theorem 11 For 120572 gt 1205721ge 0

TD (119896 120572 120573 120574 120575) sube TD (119896 1205721 120573 120574 120575) (42)

Proof Let 119891 isin TD(119896 120572 120573 120574 120575) Then consider

119867(1205721 120573 120574 120575 119891 119892) =

1 minus 1205721

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(43)

After some simple computations we have

119867(1205721 120573 120574 120575 119891 119892)

=1

1 minus 120573(1 minus

1205721

120572)[

1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

120572[

[

1 minus 120572

1 minus 120573(1198911015840 (119911)

1198921015840 (119911)minus 120573)+

120572

1 minus 120574((1199111198911015840 (119911))

1015840

1198921015840 (119911)minus 120574)]

]

=(1 minus1205721

120572)119867 (0 120573 120574 120575 119891 119892)+

1205721

120572119867 (120572 120573 120574 120575 119891 119892)

(44)

Now since119891 isin TD(119896 120572 120573 120574 120575) we have119867(120572 120573 120574 120575 119891 119892) ≺119901119896(119911) Also Theorem 8 gives us that 119867(0 120573 120574 120575 119891 119892) ≺

119901119896(119911) The use of Lemma 4 leads us to the required relation

that is 119867(1205721 120573 120574 120575 119891 119892) ≺ 119901

119896(119911) This completes the

proof

Acknowledgment

The authors would like to thank the reviewer of this paperfor hisher valuable comments on the earlier version of thispaper They would also like to acknowledge Prof Dr EhsanAli VCAWKUM for the financial support on the publicationof this article

References

[1] AW GoodmanUnivalent Functions Vol I Polygonal Publish-ing House Washington DC USA 1983

[2] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

6 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Theorem 5 A function 119891 isin A is said to be in the classTD(119896120572 120573 120574 120575) if

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) lt (1 minus 120573) (1 minus 120574) (14)

where

Φ119899(119896 120572 120573 120574 120575)

= (119896 + 1) [(1 minus 120572) (1 minus 120574) 119899 +120572 (1 minus 120573) 1198992]1003816100381610038161003816119886119899

1003816100381610038161003816

+ [(119896 + 2 minus 120573) (1 minus 120574) +120572 (119896 + 1) (120574 minus 120573) 1198991003816100381610038161003816119887119899

1003816100381610038161003816]

(15)

Proof Let us assume that relation (6) holds Now it is suffi-cient to show that

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892)minus 1

1003816100381610038161003816minus Re [119867 (120572 120573 120574 120575 119891 119892)minus 1] lt 1(16)

First we consider

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+120572

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

minus 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 minus 120572

1 minus 120573

1198911015840 (119911)

1198921015840 (119911)+

120572

1 minus 120574

(1199111198911015840 (119911))1015840

1198921015840 (119911)

minus(1 minus 120572) 120573

1 minus 120573minus

120572120574

1 minus 120574minus 1

10038161003816100381610038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) 1198911015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

+120572 (1 minus 120573) (1199111198911015840 (119911))

1015840

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] 1198921015840 (119911)

(1 minus 120573) (1 minus 120574) 1198921015840 (119911)

100381610038161003816100381610038161003816100381610038161003816

(17)

Using (1) and the series 119892(119911) = 119911 + suminfin

119899=2119887119899119911119899 in (17) we have

1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 11003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (1 + suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (1 + sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (1 + sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=2119899119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992119886119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

minus[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=2119899119887119899119911119899minus1)

(1 minus 120573) (1 minus 120574) (1 + suminfin

119899=2119899119887119899119911119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816 |119911|119899minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le100381610038161003816100381610038161003816100381610038161003816

(1 minus 120572) (1 minus 120574) (suminfin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

100381610038161003816100381610038161003816100381610038161003816

(18)

Now

1198961003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816 minus Re [119867 (120572 120573 120574 120575 119891 119892) minus 1]

le (119896 + 1)1003816100381610038161003816119867 (120572 120573 120574 120575 119891 119892) minus 1

1003816100381610038161003816

le (119896 + 1) [(1 minus 120572) (1 minus 120574) (sum

infin

119899=21198991003816100381610038161003816119886119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+120572 (1 minus 120573) (sum

infin

119899=21198992

10038161003816100381610038161198861198991003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

+[(1 minus 120574) + 120572 (120574 minus 120573)] (sum

infin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)

(1 minus 120573) (1 minus 120574) (1 minus suminfin

119899=21198991003816100381610038161003816119887119899

1003816100381610038161003816)]

(19)

4 Abstract and Applied Analysis

The last inequality is bounded above by 1 if

(119896 + 1) [(1 minus 120572) (1 minus 120574)(infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816) + 120572 (1 minus 120573)(infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816)

+ [(1 minus 120574) + 120572 (120574 minus 120573)] (infin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)]

le (1 minus 120573) (1 minus 120574)(1 minusinfin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)

(20)

Hence

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) le (1 minus 120573) (1 minus 120574) (21)

where Φ119899(119896 120572 120573 120574 120575) is given by (15) This completes the

proof

When we take 120572 = 0 119896 = 1 and 119892(119911) = 119911 in the abovetheorem we obtain the following sufficient condition for thefunctions to be in the classUK(120573) which is proved in [14]

Corollary 6 (see [14]) A function 119891 isin 119860 is said to be in theclassUK(120573) if

infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120573)

2 (22)

Corollary 7 (see [14]) A function 119891 isin 119860 is said to be in theclassUQ(120574) if

infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120574)

2 (23)

The above corollary is obtained when we take 120572 = 1 119896 =1 and 119892(119911) = 119911 in Theorem 5

Theorem 8 Let 119891 isin TD(119896 120572 120573 120574 120575) and 120572 ge 4(120573(1+3120573))Then 119891 isin KD(119896 0 120575)

Proof Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) (24)

where 119901(119911) is analytic and 119901(0) = 1 Now differentiating (24)we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 1199111199011015840 (119911) + 119901 (119911) 120595 (119911) (25)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (24) and (25) in relation(6) we obtain

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[119901 (119911) minus 120573] +

120572

1 minus 120574[1199111199011015840 (119911) + 119901 (119911) 120595 (119911) minus 120574]

=120572

1 minus 1205741199111199011015840 (119911)+[

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] 119901 (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911)

(26)

where

119861 (119911)=120572

1minus120574 119862 (119911)=

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1minus120573) (1minus120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(27)

Now since 119891 isin TD(119896 120572 120573 120574 120575) we have

119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ 119901119896(119911) (28)

Replacing 119901(119911) by 119901lowast(119911) = 119901(119911) minus 1 and 119901

119896(119911) by 119901lowast

119896(119911) =

119901119896(119911) minus 1 the above subordination is equivalent to

119861 (119911) 1199111199011015840

lowast(119911) + 119862 (119911) 119901

lowast(119911) + 119863

lowast(119911) ≺ 119901lowast

119896(119911) (29)

where119863lowast(119911) = 119862(119911) + 119863(119911) minus 1 Using Lemma 3 with 119860 = 0

we obtain

119901lowast(119911) ≺ 119901lowast

119896(119911) (30)

This implies that

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) ≺ 119901

119896(119911) (31)

Hence 119891 isin KD(119896 0 120575) This completes the proof

Corollary 9 Let 119891 isin TD(119896 120572 0 0 0) = Q120572 Then 119891 isin

KD(0 0 0) = K That is Q120572sub K 120572 ge 0

The above result is well-known inclusion proved in [11]For 119891 isin 119860 consider the following integral operator de-

fined by

119865 (119911) = 119868119898[119891] =

119898 + 1

119911119898int119911

0

119905119898minus1119891 (119905) 119889119905 119898 = 1 2 3

(32)

This operatorwas given byBernardi [18] in 1969 In particularthe operator 119868

1was considered by Libera [19] Now let us

prove the following

Abstract and Applied Analysis 5

Theorem 10 Let 119891 isin TD(119896 120572 120573 120574 120575) Then 119868119898[119891] isin

KD(119896 0 120575)

Proof Let the function 119892 be such that (6) is satisfied It caneasily be seen that according to [4] the function119866 = 119868

119898[119891] isin

CD(119896 120575) and from (32) we deduce

(1 + 119898)1198911015840

(119911) = (1 + 119898) 1198651015840

(119911) + 11991111986510158401015840 (119911) (33)

(1 + 119898) 1198921015840

(119911) = (1 + 119898)1198661015840

(119911) + 11991111986610158401015840 (119911) (34)

If we let 119901(119911) = 1198651015840(119911)1198661015840(119911) and 119902(119911) = 1(119898 + 1 +(11991111986610158401015840(119911)1198661015840)) then simple computations yield us

1198911015840 (119911)

1198921015840 (119911)=

(1 + 119898) 1198651015840 (119911) + 11991111986510158401015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

=119911119901 (119911) 1198921015840 (119911) + 119911119901 (119911) 11989210158401015840 (119911) + 1199111198921015840 (119911) 1199011015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

= 119901 (119911) + 119902 (119911) 1199111199011015840

(119911)

(35)

Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) + 1199111199011015840 (119911) 119902 (119911) = ℎ (119911) (36)

where ℎ(119911) is analytic and 119901(0) = 1 From (36) we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 120595 (119911) ℎ (119911) + 119911ℎ1015840 (119911) (37)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (36) and (37) in (6) wehave

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[ℎ (119911) minus 120573] +

120572

1 minus 120574[119911ℎ1015840 (119911) + ℎ (119911) 120595 (119911) minus 120574]

=120572

1 minus 120574119911ℎ1015840 (119911)

+ [(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] ℎ (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 119911ℎ1015840

(119911) + 119862 (119911) ℎ (119911) + 119863 (119911)

(38)

where

119861 (119911) =120572

1 minus 120574 119862 (119911) =

(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(39)

Now proceeding in the similar manner as in the proof ofTheorem 5 and using Lemma 3 with 119860 = 0 we obtain

1198911015840 (119911)

1198921015840 (119911)= ℎ (119911) ≺ 119901

119896(119911) (40)

From (36) it implies that

119901 (119911) + 1199111199011015840 (119911) 119902 (119911) ≺ 119901119896(119911) (41)

By employing Lemma 2 we immediately obtain the desiredresult

Theorem 11 For 120572 gt 1205721ge 0

TD (119896 120572 120573 120574 120575) sube TD (119896 1205721 120573 120574 120575) (42)

Proof Let 119891 isin TD(119896 120572 120573 120574 120575) Then consider

119867(1205721 120573 120574 120575 119891 119892) =

1 minus 1205721

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(43)

After some simple computations we have

119867(1205721 120573 120574 120575 119891 119892)

=1

1 minus 120573(1 minus

1205721

120572)[

1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

120572[

[

1 minus 120572

1 minus 120573(1198911015840 (119911)

1198921015840 (119911)minus 120573)+

120572

1 minus 120574((1199111198911015840 (119911))

1015840

1198921015840 (119911)minus 120574)]

]

=(1 minus1205721

120572)119867 (0 120573 120574 120575 119891 119892)+

1205721

120572119867 (120572 120573 120574 120575 119891 119892)

(44)

Now since119891 isin TD(119896 120572 120573 120574 120575) we have119867(120572 120573 120574 120575 119891 119892) ≺119901119896(119911) Also Theorem 8 gives us that 119867(0 120573 120574 120575 119891 119892) ≺

119901119896(119911) The use of Lemma 4 leads us to the required relation

that is 119867(1205721 120573 120574 120575 119891 119892) ≺ 119901

119896(119911) This completes the

proof

Acknowledgment

The authors would like to thank the reviewer of this paperfor hisher valuable comments on the earlier version of thispaper They would also like to acknowledge Prof Dr EhsanAli VCAWKUM for the financial support on the publicationof this article

References

[1] AW GoodmanUnivalent Functions Vol I Polygonal Publish-ing House Washington DC USA 1983

[2] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

6 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

The last inequality is bounded above by 1 if

(119896 + 1) [(1 minus 120572) (1 minus 120574)(infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816) + 120572 (1 minus 120573)(infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816)

+ [(1 minus 120574) + 120572 (120574 minus 120573)] (infin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)]

le (1 minus 120573) (1 minus 120574)(1 minusinfin

sum119899=2

1198991003816100381610038161003816119887119899

1003816100381610038161003816)

(20)

Hence

infin

sum119899=2

Φ119899(119896 120572 120573 120574 120575) le (1 minus 120573) (1 minus 120574) (21)

where Φ119899(119896 120572 120573 120574 120575) is given by (15) This completes the

proof

When we take 120572 = 0 119896 = 1 and 119892(119911) = 119911 in the abovetheorem we obtain the following sufficient condition for thefunctions to be in the classUK(120573) which is proved in [14]

Corollary 6 (see [14]) A function 119891 isin 119860 is said to be in theclassUK(120573) if

infin

sum119899=2

1198991003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120573)

2 (22)

Corollary 7 (see [14]) A function 119891 isin 119860 is said to be in theclassUQ(120574) if

infin

sum119899=2

11989921003816100381610038161003816119886119899

1003816100381610038161003816 le(1 minus 120574)

2 (23)

The above corollary is obtained when we take 120572 = 1 119896 =1 and 119892(119911) = 119911 in Theorem 5

Theorem 8 Let 119891 isin TD(119896 120572 120573 120574 120575) and 120572 ge 4(120573(1+3120573))Then 119891 isin KD(119896 0 120575)

Proof Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) (24)

where 119901(119911) is analytic and 119901(0) = 1 Now differentiating (24)we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 1199111199011015840 (119911) + 119901 (119911) 120595 (119911) (25)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (24) and (25) in relation(6) we obtain

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[119901 (119911) minus 120573] +

120572

1 minus 120574[1199111199011015840 (119911) + 119901 (119911) 120595 (119911) minus 120574]

=120572

1 minus 1205741199111199011015840 (119911)+[

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] 119901 (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911)

(26)

where

119861 (119911)=120572

1minus120574 119862 (119911)=

(1minus120572) (1minus120574)+120572 (1minus120573)120595 (119911)

(1minus120573) (1minus120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(27)

Now since 119891 isin TD(119896 120572 120573 120574 120575) we have

119861 (119911) 1199111199011015840

(119911) + 119862 (119911) 119901 (119911) + 119863 (119911) ≺ 119901119896(119911) (28)

Replacing 119901(119911) by 119901lowast(119911) = 119901(119911) minus 1 and 119901

119896(119911) by 119901lowast

119896(119911) =

119901119896(119911) minus 1 the above subordination is equivalent to

119861 (119911) 1199111199011015840

lowast(119911) + 119862 (119911) 119901

lowast(119911) + 119863

lowast(119911) ≺ 119901lowast

119896(119911) (29)

where119863lowast(119911) = 119862(119911) + 119863(119911) minus 1 Using Lemma 3 with 119860 = 0

we obtain

119901lowast(119911) ≺ 119901lowast

119896(119911) (30)

This implies that

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) ≺ 119901

119896(119911) (31)

Hence 119891 isin KD(119896 0 120575) This completes the proof

Corollary 9 Let 119891 isin TD(119896 120572 0 0 0) = Q120572 Then 119891 isin

KD(0 0 0) = K That is Q120572sub K 120572 ge 0

The above result is well-known inclusion proved in [11]For 119891 isin 119860 consider the following integral operator de-

fined by

119865 (119911) = 119868119898[119891] =

119898 + 1

119911119898int119911

0

119905119898minus1119891 (119905) 119889119905 119898 = 1 2 3

(32)

This operatorwas given byBernardi [18] in 1969 In particularthe operator 119868

1was considered by Libera [19] Now let us

prove the following

Abstract and Applied Analysis 5

Theorem 10 Let 119891 isin TD(119896 120572 120573 120574 120575) Then 119868119898[119891] isin

KD(119896 0 120575)

Proof Let the function 119892 be such that (6) is satisfied It caneasily be seen that according to [4] the function119866 = 119868

119898[119891] isin

CD(119896 120575) and from (32) we deduce

(1 + 119898)1198911015840

(119911) = (1 + 119898) 1198651015840

(119911) + 11991111986510158401015840 (119911) (33)

(1 + 119898) 1198921015840

(119911) = (1 + 119898)1198661015840

(119911) + 11991111986610158401015840 (119911) (34)

If we let 119901(119911) = 1198651015840(119911)1198661015840(119911) and 119902(119911) = 1(119898 + 1 +(11991111986610158401015840(119911)1198661015840)) then simple computations yield us

1198911015840 (119911)

1198921015840 (119911)=

(1 + 119898) 1198651015840 (119911) + 11991111986510158401015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

=119911119901 (119911) 1198921015840 (119911) + 119911119901 (119911) 11989210158401015840 (119911) + 1199111198921015840 (119911) 1199011015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

= 119901 (119911) + 119902 (119911) 1199111199011015840

(119911)

(35)

Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) + 1199111199011015840 (119911) 119902 (119911) = ℎ (119911) (36)

where ℎ(119911) is analytic and 119901(0) = 1 From (36) we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 120595 (119911) ℎ (119911) + 119911ℎ1015840 (119911) (37)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (36) and (37) in (6) wehave

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[ℎ (119911) minus 120573] +

120572

1 minus 120574[119911ℎ1015840 (119911) + ℎ (119911) 120595 (119911) minus 120574]

=120572

1 minus 120574119911ℎ1015840 (119911)

+ [(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] ℎ (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 119911ℎ1015840

(119911) + 119862 (119911) ℎ (119911) + 119863 (119911)

(38)

where

119861 (119911) =120572

1 minus 120574 119862 (119911) =

(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(39)

Now proceeding in the similar manner as in the proof ofTheorem 5 and using Lemma 3 with 119860 = 0 we obtain

1198911015840 (119911)

1198921015840 (119911)= ℎ (119911) ≺ 119901

119896(119911) (40)

From (36) it implies that

119901 (119911) + 1199111199011015840 (119911) 119902 (119911) ≺ 119901119896(119911) (41)

By employing Lemma 2 we immediately obtain the desiredresult

Theorem 11 For 120572 gt 1205721ge 0

TD (119896 120572 120573 120574 120575) sube TD (119896 1205721 120573 120574 120575) (42)

Proof Let 119891 isin TD(119896 120572 120573 120574 120575) Then consider

119867(1205721 120573 120574 120575 119891 119892) =

1 minus 1205721

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(43)

After some simple computations we have

119867(1205721 120573 120574 120575 119891 119892)

=1

1 minus 120573(1 minus

1205721

120572)[

1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

120572[

[

1 minus 120572

1 minus 120573(1198911015840 (119911)

1198921015840 (119911)minus 120573)+

120572

1 minus 120574((1199111198911015840 (119911))

1015840

1198921015840 (119911)minus 120574)]

]

=(1 minus1205721

120572)119867 (0 120573 120574 120575 119891 119892)+

1205721

120572119867 (120572 120573 120574 120575 119891 119892)

(44)

Now since119891 isin TD(119896 120572 120573 120574 120575) we have119867(120572 120573 120574 120575 119891 119892) ≺119901119896(119911) Also Theorem 8 gives us that 119867(0 120573 120574 120575 119891 119892) ≺

119901119896(119911) The use of Lemma 4 leads us to the required relation

that is 119867(1205721 120573 120574 120575 119891 119892) ≺ 119901

119896(119911) This completes the

proof

Acknowledgment

The authors would like to thank the reviewer of this paperfor hisher valuable comments on the earlier version of thispaper They would also like to acknowledge Prof Dr EhsanAli VCAWKUM for the financial support on the publicationof this article

References

[1] AW GoodmanUnivalent Functions Vol I Polygonal Publish-ing House Washington DC USA 1983

[2] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

6 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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

Abstract and Applied Analysis 5

Theorem 10 Let 119891 isin TD(119896 120572 120573 120574 120575) Then 119868119898[119891] isin

KD(119896 0 120575)

Proof Let the function 119892 be such that (6) is satisfied It caneasily be seen that according to [4] the function119866 = 119868

119898[119891] isin

CD(119896 120575) and from (32) we deduce

(1 + 119898)1198911015840

(119911) = (1 + 119898) 1198651015840

(119911) + 11991111986510158401015840 (119911) (33)

(1 + 119898) 1198921015840

(119911) = (1 + 119898)1198661015840

(119911) + 11991111986610158401015840 (119911) (34)

If we let 119901(119911) = 1198651015840(119911)1198661015840(119911) and 119902(119911) = 1(119898 + 1 +(11991111986610158401015840(119911)1198661015840)) then simple computations yield us

1198911015840 (119911)

1198921015840 (119911)=

(1 + 119898) 1198651015840 (119911) + 11991111986510158401015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

=119911119901 (119911) 1198921015840 (119911) + 119911119901 (119911) 11989210158401015840 (119911) + 1199111198921015840 (119911) 1199011015840 (119911)

(1 + 119898)1198661015840 (119911) + 11991111986610158401015840 (119911)

= 119901 (119911) + 119902 (119911) 1199111199011015840

(119911)

(35)

Let

1198911015840 (119911)

1198921015840 (119911)= 119901 (119911) + 1199111199011015840 (119911) 119902 (119911) = ℎ (119911) (36)

where ℎ(119911) is analytic and 119901(0) = 1 From (36) we have

(1199111198911015840 (119911))1015840

1198921015840 (119911)= 120595 (119911) ℎ (119911) + 119911ℎ1015840 (119911) (37)

where 120595(119911) = (1199111198921015840(119911))1015840

1198921015840(119911) Using (36) and (37) in (6) wehave

119867(120572 120573 120574 120575 119891 (119911))

=1 minus 120572

1 minus 120573[ℎ (119911) minus 120573] +

120572

1 minus 120574[119911ℎ1015840 (119911) + ℎ (119911) 120595 (119911) minus 120574]

=120572

1 minus 120574119911ℎ1015840 (119911)

+ [(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)] ℎ (119911)

minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

= 119861 (119911) 119911ℎ1015840

(119911) + 119862 (119911) ℎ (119911) + 119863 (119911)

(38)

where

119861 (119911) =120572

1 minus 120574 119862 (119911) =

(1 minus 120572) (1 minus 120574) + 120572 (1 minus 120573)120595 (119911)

(1 minus 120573) (1 minus 120574)

119863 (119911) = minus120573 (1 minus 120572) (1 minus 120574) + 120572120574 (1 minus 120573)

(1 minus 120573) (1 minus 120574)

(39)

Now proceeding in the similar manner as in the proof ofTheorem 5 and using Lemma 3 with 119860 = 0 we obtain

1198911015840 (119911)

1198921015840 (119911)= ℎ (119911) ≺ 119901

119896(119911) (40)

From (36) it implies that

119901 (119911) + 1199111199011015840 (119911) 119902 (119911) ≺ 119901119896(119911) (41)

By employing Lemma 2 we immediately obtain the desiredresult

Theorem 11 For 120572 gt 1205721ge 0

TD (119896 120572 120573 120574 120575) sube TD (119896 1205721 120573 120574 120575) (42)

Proof Let 119891 isin TD(119896 120572 120573 120574 120575) Then consider

119867(1205721 120573 120574 120575 119891 119892) =

1 minus 1205721

1 minus 120573[1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

1 minus 120574[

[

(1199111198911015840 (119911))1015840

1198921015840 (119911)minus 120574]

]

(43)

After some simple computations we have

119867(1205721 120573 120574 120575 119891 119892)

=1

1 minus 120573(1 minus

1205721

120572)[

1198911015840 (119911)

1198921015840 (119911)minus 120573]

+1205721

120572[

[

1 minus 120572

1 minus 120573(1198911015840 (119911)

1198921015840 (119911)minus 120573)+

120572

1 minus 120574((1199111198911015840 (119911))

1015840

1198921015840 (119911)minus 120574)]

]

=(1 minus1205721

120572)119867 (0 120573 120574 120575 119891 119892)+

1205721

120572119867 (120572 120573 120574 120575 119891 119892)

(44)

Now since119891 isin TD(119896 120572 120573 120574 120575) we have119867(120572 120573 120574 120575 119891 119892) ≺119901119896(119911) Also Theorem 8 gives us that 119867(0 120573 120574 120575 119891 119892) ≺

119901119896(119911) The use of Lemma 4 leads us to the required relation

that is 119867(1205721 120573 120574 120575 119891 119892) ≺ 119901

119896(119911) This completes the

proof

Acknowledgment

The authors would like to thank the reviewer of this paperfor hisher valuable comments on the earlier version of thispaper They would also like to acknowledge Prof Dr EhsanAli VCAWKUM for the financial support on the publicationof this article

References

[1] AW GoodmanUnivalent Functions Vol I Polygonal Publish-ing House Washington DC USA 1983

[2] S Kanas and A Wisniowska ldquoConic regions and 119896-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999

6 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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 Abstract and Applied Analysis

[3] S Kanas andAWisniowska ldquoConic domains and starlike func-tionsrdquo Revue Roumaine de Mathematiques Pures et Appliqueesvol 45 no 4 pp 647ndash657 2000

[4] M Acu ldquoOn a subclass of 119899-uniformly close to convex func-tionsrdquo General Mathematics vol 14 no 1 pp 55ndash64 2006

[5] E Aqlan J M Jahangiri and S R Kulkarni ldquoNew classes of 119896-uniformly convex and starlike functionsrdquo Tamkang Journal ofMathematics vol 35 no 3 pp 1ndash7 2004

[6] S Kanas ldquoAlternative characterization of the class 119896-119880119862119881 andrelated classes of univalent functionsrdquo Serdica MathematicalJournal vol 25 no 4 pp 341ndash350 1999

[7] S Kanas andHM Srivastava ldquoLinear operators associatedwith119896-uniformly convex functionsrdquo Integral Transforms and SpecialFunctions vol 9 no 2 pp 121ndash132 2000

[8] S Shams S R Kulkarni and J M Jahangiri ldquoClasses of uni-formly starlike and convex functionsrdquo International Journal ofMathematics and Mathematical Sciences vol 2004 no 55 pp2959ndash2961 2004

[9] H M Srivastava S-H Li and H Tang ldquoCertain classes of 119896-uniformly close-to-convex functions and other related func-tions defined by using the Dziok-Srivastava operatorrdquo Bulletinof Mathematical Analysis and Applications vol 1 no 3 pp 49ndash63 2009

[10] K I Noor ldquoOn a generalization of uniformly convex and relatedfunctionsrdquo Computers ampMathematics with Applications vol 61no 1 pp 117ndash125 2011

[11] K I Noor and FM Al-Oboudi ldquoAlpha-quasi-convex univalentfunctionsrdquo Caribbean Journal of Mathematics vol 3 pp 1ndash81984

[12] K I Noor and S N Malik ldquoOn generalized bounded Mocanuvariation associated with conic domainrdquo Mathematical andComputer Modelling vol 55 no 3-4 pp 844ndash852 2012

[13] K I Noor M Arif and W Ul-Haq ldquoOn 119896-uniformly close-to-convex functions of complex orderrdquo Applied Mathematics andComputation vol 215 no 2 pp 629ndash635 2009

[14] KG Subramanian TV Sudharsan andH Silverman ldquoOnuni-formly close-to-convex functions and uniformly quasiconvexfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2003 no 48 pp 3053ndash3058 2003

[15] R J Libera ldquoSome radius of convexity problemsrdquo Duke Mathe-matical Journal vol 31 pp 143ndash158 1964

[16] K I Noor ldquoOn quasi-convex functions and related topicsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2 pp 241ndash258 1987

[17] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[18] S D Bernardi ldquoConvex and starlike univalent functionsrdquo Tran-sactions of the AmericanMathematical Society vol 135 pp 429ndash446 1969

[19] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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