actamathscientia32b(3)(2012)929-941cho

7/31/2019 ActaMathScientia32B(3)(2012)929-941Cho

1/13

Acta Mathematica Scientia 2012,32B(3):929941

http://actams.wipm.ac.cn

SANDWICH-TYPE THEOREMS FOR

MEROMORPHIC MULTIVALENT FUNCTIONS

ASSOCIATED WITH THE LIU-SRIVASTAVA

OPERATOR

Nak Eun Cho

Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

E-mail: [email protected]

Abstract The purpose of this article is to obtain some subordination and superordi-

nation preserving properties of meromorphic multivalent functions in the punctured open

unit disk associated with the Liu-Srivastava operator. The sandwich-type results for these

meromorphic multivalent functions are also considered.

Key words Subordination; superordination; meromorphic multivalent function; Liu-

Srivastava operator; sandwich-type result

2000 MR Subject Classification 30C80; 30C45; 30D30

1 Introduction

Let H = H(U) denote the class of analytic functions in the open unit disk

U = {z C : |z| < 1}.

For n N = {1, 2, } and a C, let

H[a, n] = {f H : f(z) = a + anzn + an+1z

n+1 + }.

Let f and F be members of H. The function f is said to be subordinate to F, or F is

said to be superordinate to f, if there exists a function w analytic in U, with w(0) = 0 and

|w(z)| < 1 (z U), such that f(z) = F(w(z)) (z U).

In such a case, we write

f F (z U) or f(z) F(z) (z U).

Received April 15, 2010; revised November 23, 2010. This research was supported by the Basic Science Re-

search Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education,

Science and Technology (2010-0017111).


2/13

930 ACTA MATHEMATICA SCIENTIA Vol.32 Ser.B

If the function F is univalent in U, then we have (cf. [1])

f F (z U) f(0) = F(0) and f(U) F(U).

Definition 1 [1] Let : C2 C and let h be univalent in U. If p is analytic in U and

satisfies the differential subordination:

(p(z), zp(z)) h(z) (z U), (1.1)

then p is called a solution of the differential subordination. The univalent function q is called a

dominant of the solutions of the differential subordination, or more simply a dominant, if p q

for all p satisfying (1.1). A dominant q that satisfies q q for all dominants q of (1.1) is said

to be the best dominant.

Definition 2 [2] Let : C2 C and let h be analytic in U. If p and (p(z), zp(z)) are

univalent in U and satisfy the differential superordination:

h(z) (p(z), zp(z)) (z U), (1.2)

then p is called a solution of the differential superordination. An analytic function q is called asubordinant of the solutions of the differential superordination, or more simply a subordinant if

q p for all p satisfying (1.2). A univalent subordinant q that satisfies q q for all subordinants

q of (1.2) is said to be the best subordinant.

Definition 3 [2] Denote by Q the class of functions f that are analytic and injective on

U\E(f), where

E(f) =

U : limz

f(z) =

,

and are such that

f() = 0 ( U\E(f)).

Let p denote the class of functions of the form

f(z) = zp +

k=1

akzkp (p N = {1, 2, }), (1.3)

which are analytic and p-valent in the punctured unit disk D = U\{0} with the following

additional condition:

limz0

(zpf(z)) = 0 (z D). (1.4)

For functions f p given by (1.3), and g p given by

g(z) = zp +

k=1

bkzkp (p N),

define the Hadamard product (or convolution) of f and g by

(f g)(z) := zp +k=1

akbkzkp.


3/13

No.3 Nak Eun Cho: SANDWICH-TYPE THEOREMS 931

For parameters j C (j = 1, , l) and j C\Z

0 (Z

0 := 0, 1, 2, ; j = 1, , m),

the generalized hypergeometric function lFm(1, , l; 1, , m; z) is defined by the follow-

ing infinite series (cf. [35]):

lFm(1, , l; 1, , m; z) :=k=0

(1)k (l)k(1)k (m)k

zk

k!

(l m + 1; l, m N0 := N {0}; z U),

where ()k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma

function) by

()k :=(+ k)

()=

1 if k = 0 and C\{0},(+ 1) (+ k 1) ifk N and C.

Corresponding to a function Fp(1, , l; 1, , m; z) defined by

Fp(1, , l; 1, , m; z) := z

p

lFm(1, , l; 1, , m; z),

the Liu-Srivastava operator Hl,mp (1, , lq; 1, , m) : p p is defined by the fol-

lowing Hadamard product (or convolution):

Hl,mp (1, , l; 1, , m)f(z) := Fp(1, , l; 1, , m; z) f(z).

We observe that, for a function f given by (1.3), we have

Hl,mp (1, , l; 1, , m)f(z) =1

zp+

k=1

(1)k (l)k(1)k (m)k

akk!

zkp.

To make the notation simple, we write

Hl,mp (1) := Hl,mp (1, , l; 1, , m) (l m + 1; l, m N0). (1.5)

The operator Hl,mp (1) was defined and studied by Liu and Srivastava [6]. We also note that

the definition of operator Hl,mp (1) was motivated essentially by Dziok and Srivastava [3].

Some interesting developments involving the Dziok-Srivastava operator were considered by (for

example) Dziok and Srivastava ([7], also see [812]) and Liu and Srivastava [5, 13, 14].

Using the principle of subordination, various subordination theorems involving certain in-

tegral operators for analytic functions in U were established in Bulboaca [15], Miller et al [16]

and Owa and Srivastava [17]. Recently, Miller and Mocanu [2] also considered differential su-

perordinations, as the dual problem of differential subordinations (see, also [18]). It should be

also remarked that, in recent years, several authors obtained many interesting results involving

various linear and nonlinear operators associated with differential subordination and superordi-

nation [1925]. In this article, we investigate the subordination and superordination preserving

properties of the linear operator Hl,mp (1) defined by (1.5) with the sandwich-type theorems.


4/13


2 A Set of Lemmas

The following lemmas will be required in our present investigation.

Lemma 1 [1] Let p Q with p(0) = a and let

q(z) = a + anzn +

be analytic in U withq(z) a and n 1.

If q is not subordinate to p, then there exist points

z0 = r0ei U and 0 U\E(f),

for which

q(Ur0) p(U), q(z0) = p(0) and z0q(z0) = m0p

(0) (m n).

A function L(z, t) defined on U [0, ) is the subordination chain (or Lowner chain) if

L(, t) is analytic and univalent in U for all t [0, ), L(z, ) is continuously differentiable on

[0, ) for all z U and L(z, s) L(z, t) (z U; 0 s < t).

Lemma 2 [2] Let q H[a, 1] and : C2 C. Also set

(q(z), zq (z)) h(z) (z U).

If L(z, t) = (q(z),tzq(z)) is a subordination chain and p H[a, 1] Q, then,

h(z) (p(z), zp(z)) (z U).

implies that

q(z) p(z) (z U).

Furthermore, if (q(z), zp

(z)) = h(z) has a univalent solution q Q, then q is the bestsubordinant.

Lemma 3 [26] Suppose that the function H : C2 C satisfies the following condition:

R{H(is, t)} 0

for all real s and

t n(1 + s2)/2 (n N).

If the function p(z) = 1 + pnzn + is analytic in U and

R{H(p(z), zp(z))} > 0 (z U),

then,

R{p(z)} > 0 (z U).

Lemma 4 [27] Let , C with = 0 and let h H(U) with h(0) = c. If

R{h(z) + } > 0 (z U),


5/13


then, the solution of the differential equation

q(z) +zq (z)

q(z) + = h(z) (z U; q(0) = c)

is analytic in U and satisfies the inequality

R{q(z) + } > 0 (z U).

Lemma 5 [28] The function L(z, t) = a1(t)z + with a1(t) = 0 and limt

|a1(t)| =

is a subordination chain if and only if

R

zL(z,t)

z

L(z,t)t

> 0 (z U; 0 t < ).

3 Main Results

We begin with proving the following subordination theorem involving the operator Hl,mp (1)

defined by (1.5).

Theorem 1 Let f, g p. Suppose that

R

1 +

z(z)

(z)

> (3.1)

(z) :=

p

pzpHl,mp (1 + 1)g(z) +

pzpHl,mp (1)g(z); 1 > 0; 0 < p; z U

,

where

=(p )2 + p221 |(p )

2 p221|

4p(p )1(0 < 1/2). (3.2)

Then, the following subordination relation

p

pzpHl,mp (1 + 1)g(z) +

pzpHl,mp (1)f(z) (z) (z U) (3.3)

implies thatzpHl,mp (1)f(z) z

pHl,mp (1)g(z) (z U). (3.4)

Moreover, the function zpHl,mp (1)g(z) is the best dominant.

Proof Let us define the functions F and G, respectively, by

F(z) := zpHl,mp (1)f(z) and G(z) := zpHl,mp (1)g(z). (3.5)

We first show that, if the function q is defined by

q(z) := 1 +zG(z)

G(z)(z U), (3.6)

then,

R{q(z)} > 0 (z U).

Taking the logarithmic differentiation on both sides of the second equation in (3.5) and

using the equation

z(Hl,mp (1)g(z)) = 1H

l,mp (1 + 1)g(z) (1 + p)H

l,mp (1)g(z),


6/13


we obtain

p1(z) = p1G(z) + (p )zG(z). (3.7)

Now, by differentiating both sides of (3.7), we obtain the relationship:

1 +z(z)

(z)= 1 +

zG(z)

G(z)+

zq (z)

q(z) + p1/(p )

= q(z) + zq

(z)q(z) + p1/(p )

h(z). (3.8)

We also note from (3.1) that

R

h(z) +

p1p

> 0 (z U),

and so by Lemma 4, we conclude that the differential equation (3.8) has a solution q H(U)

with

q(0) = h(0) = 1.

Let us put

H(u, v) = u +v

u + p1/(p ) + , (3.9)

where is given by (3.2). From (3.1), (3.8), and (3.9), we obtain

R{H(q(z), zq (z))} > 0 (z U).

Now, we proceed to show that

R{H(is, t)} 0

s R; t

1

2(1 + s2)

. (3.10)

Indeed, From (3.9), we have

R{H(is, t)} = R

is + tis + p1/(p )

+

=tp1/p

|p1/(p ) + is|2+

E(s)

2|p1/(p ) + is|2, (3.11)

where

E(s) :=

p1

p 2

s2

p1p

2

p1p

1

. (3.12)

For given by (3.2), we can prove easily that the expression E(s) given by (3.12) is greater

than or equal to zero. Hence, from (3.9), we see that (3.10) holds true. Thus, using Lemma 3,

we conclude that

R{q(z)} > 0 (z U).

Moreover, we see that the condition:

G(0) = 0

is satisfied. Hence, the function G defined by (3.5) is convex in U.


7/13


Next, we prove that the subordination condition (3.3) implies that

F(z) G(z) (z U) (3.13)

for the functions F and G defined by (3.5). Without loss of generality, we can assume that G

is analytic and univalent on U and

G

() = 0 ( U).

For this purpose, we consider the function L(z, t) given by

L(z, t) := G(z) +(p )(1 + t)

p1zG(z) (z U; 0 t < ; 0 < 1).

We note that

L(z, t)

z

z=0

= G(0)

p1 + (p )(1 + t)

1

= 0 (0 t < ; 1 > 0; 0 < 1).

This shows that the function

L(z, t) = a1(t)z +

satisfies the condition

a1(t) = 0 (0 t < ).

Furthermore, we have

R

z L(z, t)/z

L(z, t)/t

= R

p1

p + (1 + t)

1 +

zG(z)

G(z)

> 0.

Therefore, by virtue of Lemma 5, L(z, t) is a subordination chain. We observe from the

definition of subordination chain that

L(, t) L(U, 0) = (U) ( U; 0 t < ).

Now, suppose that F is not subordinate to G, then by Lemma 1, there exists points z0 U

and 0 U, such that

F(z0) = G(0) and z0F(z0) = (1 + t)0G(0) (0 t < ).

Hence, we have

L(0, t) = G(0) +(p )(1 + t)

p10G

(0) = F(z0) +p

p1z0F

(z0)

=p

pz0(H

l,mp (1 + 1)f(z0)) +

pz0(H

l,mp (1)f(z0)) (U),

by virtue of the subordination condition (3.3). This contradicts the above observation L(0, t)

(U). Therefore, the subordination condition (3.3) must imply the subordination given by

(3.13). Considering F(z) = G(z), we see that the function G is the best dominant. This

evidently completes the proof of Theorem 1.


8/13


We next provide a dual problem of Theorem 1, in the sense that the subordinations are

replaced by superordinations.

Theorem 2 Let f, g p. Suppose that

R

1 +

z(z)

(z)

>

(z) := p

zpHl,mp (1 + 1)g(z) + p

zpHl,mp (1)g(z); 1 > 0; 0 < p; z U

,

where is given by (3.2), and

p

zpHl,mp (1 + 1)f(z) +

pzpHl,mp (1)f(z)

is univalent in U and zpHl,mp (1)f(z) H[1, 1]Q. Then, the following superordination relation

(z) p

zpHl,mp (1 + 1)f(z) +

pzpHl,mp (1)f(z) (z U) (3.14)

implies that

zp

Hl,mp (1)g(z) z

p

Hl,mp (1)f(z) (z U).

Moreover, the function zpHl,mp (1)g(z) is the best subordinant.

Proof The first part of the proof is similar to that of Theorem 1 and so we will use the

same notation as in the proof of Theorem 1.

Now, let us define the functions F and G by (3.5). We first note that, if the function q is

defined by (3.6), using (3.7), then we obtain

(z) = G(z) +p

p1zG(z) =: (G(z), zG(z)). (3.15)

Then using the same method as in the proof of Theorem 1, we can prove that

R{q(z)} > 0 (z U),

that is, G defined by (3.5) is convex(univalent) in U.

Next, we prove that the subordination condition (3.14) implies that

G(z) F(z) (z U) (3.16)

for the functions F and G defined by (3.5). Now, consider the function L(z, t) defined by

L(z, t) := G(z) +(p )t

p1zG(z) (z U; 0 t < ).

As G is convex and p1/(p ) > 0, we can prove easily that L(z, t) is a subordination

chain as in the proof of Theorem 1. Therefore, according to Lemma 2, we conclude that the

superordination condition (3.14) must imply the superordination given by (3.16). Furthermore,

as the differential equation (3.15) has the univalent solution G, it is the best subordinant of the

given differential superordination. Therefore, we complete the proof of Theorem 2.

If we combine Theorems 1 and 2, then, we obtain the following sandwich-type theorem.


9/13


Theorem 3 Let f, gk p (k = 1, 2). Suppose that

R

1 +

zk(z)

k(z)

> (3.17)

k(z) : =p

pzpHl,mp (1 + 1)gk(z) +

pzpHl,mp (1)gk(z);

k = 1, 2; 1 > 0; 0 < p; z U

,

where is given by (3.2), and

p

pzpHl,mp (1 + 1)f(z) +

pzpHl,mp (1)f(z)

is univalent in U and zpHl,mp (1)f(z) H[1, 1] Q. Then, the following relation

1(z) p


pzpHl,mp (1)f(z) 2(z) (z U)

implies that

zpHl,mp (1)g1(z) zpHl,mp (1)f(z) z

pHl,mp (1)g2(z) (z U).

Moreover, the functions zpHl,mp (1)g1(z) and zpHl,mp (1)g2(z) are the best subordinant

and the best dominant, respectively.

The assumption of Theorem 3, that the functions

p


pzpHl,mp (1)f(z) and z

pHl,mp (1)f(z)

need to be univalent in U, may be replaced by another condition in the following result.

Corollary 1 Let f, gk p (k = 1, 2). Suppose that condition (3.17) is satisfied and

R

1 + z

(z)(z)

> (3.18)

(z) :=

p


pzpHl,mp (1)f(z); 1 > 0; 0 < p; z U

,

where is given by (3.2). Then, the following relation

1(z) p


pzpHl,mp (1)f(z) 2(z) (z U)

implies that





Proof To prove Corollary 1, we have to show that condition (3.18) implies the univalence

of (z) and

F(z) := zpHl,mp (1)f(z).


10/13


As 0 < 1/2 from Theorem 1, condition (3.18) means that is a close-to-convex function

in U (see [29]) and hence is univalent in U. Furthermore, using the same techniques as in

the proof of Theorem 1, we can prove the convexity(univalence) of F and so the details may be

omitted here. Therefore, by applying Theorem 3, we obtain Corollary 1.

Taking q = s + 1, 1 = = p, i = i (i = 2, 3, , s), s+1 = 1, and = 0 in Theorem

3, we have the following result.

Corollary 2 Let f, g p. Suppose that

R

1 +

zk(z)

k(z)

>

1

2p(z U; k(z) := z

p(zg k(z) + (1 + p)gk(z)); k = 1, 2) ,

and zp(zf(z) + (1 + p)f(z)) is univalent in U, and zpf(z) H[1, 1] Q. Then,

zp(zg 1(z) + (1 + p)g1(z)) zp(zf(z) + (1 + p)f(z)) zp(zg 2(z) + (1 + p)g2(z)) (z U)

implies that

zpg1(z) zpf(z) zpg2(z) (z U).

Moreover, the functions zpg1(z) and zpg2(z) are the best subordinant and the best domi-

nant, respectively.The proof of Theorem 4 below is similar to that of Theorem 3 using (1.6), and so it is

omitted.

Theorem 4 Let f, gk p (k = 1, 2). Suppose that

R

1 +

zk(z)

k(z)

>

k(z) : =

p

pzp+1Hl,mp (1 + 1)gk(z) +

pzp+1Hl,mp (1)gk(z);

k = 1, 2; 1 > (p )/p; 0 < p; z U,where

=(p )2 + (p(1 1) + )2 |(p )2 (p(1 1) + )2|

4(p )(p(1 1) + ),

andp

pzp+1Hl,mp (1 + 1)f(z) +

pzp+1Hl,mp (1)f(z)

is univalent in U and zp+1(Hl,mp (1)f(z)) H[0, 1] Q. Then, the following relation

1(z) p

pzp+1Hl,mp (1 + 1)f(z) +

pzp+1Hl,mp (1)f(z) 2(z) (z U)

implies that






11/13


Next, we consider the integral operator F ( > 0) defined by (cf. [3032])

Fu(f)(z) :=

z+p

z0

t+p1f(t)dt (f p; > 0) (3.19)

Now, we obtain the following result involving the integral operator defined by (3.19).

Theorem 5 Let f, gk p(k = 1, 2). Suppose also that

R

1 +

zk(z)

k(z)

>

k(z) := z

pHl,mp (1)gk(z); k = 1, 2; z U

, (3.20)

where

=1 + 2 |1 2|

4( > 0), (3.21)

and zpHl,mp (1)f(z) is univalent in U and zpHl,mp (1)F(f)(z) H[1, 1] Q. Then, the follow-

ing relation

1(z) zpHl,mp (1)f(z) 2(z) (z U)

implies that

zp

Hl,m

p (1)F(g1)(z) zp

Hl,m

p (1)F(f)(z) zp

Hl,m

p (1)F(g2)(z) (z U).

Moreover, the functions zpHl,mp (1)F(g1)(z) and zpHl,mp (1)F(g2)(z) are the best sub-

ordinant and the best dominant, respectively.

Proof Let us define the functions F and Gk (k = 1, 2) by

F(z) := zpHl,mp (1)F(f)(z) and Gk(z) := zpHl,mp (1)F(gk)(z),

respectively. Without loss of generality, as in the proof of Theorem 1, we can assume that Gk

is analytic and univalent on U and

Gk() = 0 ( U).

From the definition of the integral operator F defined by (3.19), we obtain

z(Hl,mp (1)F(f)(z)) = Hl,mp (1)f(z) ( + p)H

l,mp (1)F(f)(z). (3.22)

Then, from (3.20) and (3.22), we have

k(z) = Gk(z) + zG

k(z). (3.23)

Setting

qk(z) = 1 +zGk(z)

Gk(z)(k = 1, 2; z U)

and differentiating both sides of (3.23), we obtain

1 +zk(z)

k(z)= qk(z) +

zq k(z)

qk(z) + .

The remaining part of the proof is similar to that of Theorem 1 and so is omitted the proof

involved.


12/13


Using the same methods as in the proof of Corollary 1, we have the following result.

Corollary 3 Let f, gk p(k = 1, 2). Suppose that condition (3.20) is satisfied and

R

1 +

z (z)

(z)

>

(z) := zpHl,mp (1)f(z) : z U

,

where is given by (3.21). Then,

1(z) zpHl,mp (1)f(z) 2(z) (z U)

implies that

zpHl,mp (1)F(g1)(z) zpHl,mp (1)F(f)(z) z

pHl,mp (1)F(g2)(z) (z U).

Moreover, the functions zpHl,mp (1)F(g1)(z) and zpHl,mp (1)F(g2)(z) are the best sub-

ordinant and the best dominant, respectively.

Taking q = s + 1, 1 = 1 = p, i = i (i = 2, 3, , s), and s+1 = 1 in Theorem 5, we

have the following result.

Corollary 4 Let f, gk p (k = 1, 2). Suppose also that

R

1 + z

k(z)k(z)

> (z U; k(z) := zpgk(z); k = 1, 2) ,

where is given by (3.21), and zpf(z) is univalent in U and zp(F(f)(z)) H[1, 1] Q, then,

zpg1(z) zpf(z) zpg2(z) (z U)

implies that

zp(F(g1)(z)) zp(F(f)(z)) z

p(F(g2)(z)) (z U).

Moreover, the functions zp(F(g1)(z)) and zp(F(g2)(z)) are the best subordinant and the

best dominant, respectively.

References

[1] Miller S S, Mocanu P T. Differential Subordination, Theory and Applications. New York, Basel: Marcel

Dekker Inc, 2000

[2] Miller S S, Mocanu P T. Subordinants of differential superordinations. Complex Var Theory Appl, 2003,

48: 815826

[3] Dziok J, Srivastava H M. Classes of analytic functions associated with the generalized hypergeometric

function. Appl Math Comput, 1999, 103: 113

[4] Dziok J, Srivastava H M. Certain subclasses of analytic functions associated with the generalized hyper-

geometric function. Integral Transforms Spec Funct, 2003, 14: 718

[5] Liu J L, Srivastava H M. Certain properties of the Dziok-Srivastava operator. Appl Math Comput, 2004,

159: 485493

[6] Liu J L, Srivastava H M. Classes of meromorphically multivalent functions associated with the generalized

hypergeometric function. Math Comput Modelling, 2004, 39: 2134

[7] Dziok J, Srivastava H M. Some subclasses of analytic functions with fixed argument of coefficients associ-

ated with the generalized hypergeometric function. Adv Stud Contemp Math, 2002, 5: 115125

[8] Liu J L. On subordinations for certain analytic functions associated with the Dziok-Srivastava linear

operator. Taiwanese J Math, 2009, 13: 349357

[9] Patel J, Mishra A K, Srivastava H M. Classes of multivalent analytic functions involving the Dziok-

Srivastava operator. Comput Math Appl, 2007, 54: 599616


13/13


[10] Ramachandran C, Shanmugam T N, Srivastava H M, Swaminathan A. A unified class of k-uniformly

convex functions defined by the Dziok-Srivastava operator. Appl Math Comput, 2007, 190: 16271636

[11] Srivastava H M, Owa S. Some characterizations and distortions theorems involving fractional calculus,

generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of

analytic functions. Nagoya Math J, 1987, 106: 128

[12] Owa S, Srivastava H M. Univalent and starlike generalized hypergeometric functions. Canad J Math, 1987,

39: 10571077

[13] Liu J L, Srivastava H M. A linear operator and associated families of meromorphically multivalent func-

tions. J Math Anal Appl, 2001, 259: 566581

[14] Liu J L, Srivastava H M. Certain properties of the Dziok-Srivastava operator. Appl Math Comput, 2004,

159: 485493

[15] Bulboaca T. Integral operators that preserve the subordination. Bull Korean Math Soc, 1997, 32: 627636

[16] Miller S S, Mocanu P T, Reade M O. Subordination-preserving integral operators. Trans Amer Math Soc,

1984, 283: 605615

[17] Owa S, Srivastava H M. Some subordination theorems involving a certain family of integral operators.

Integral Transforms Spec Funct, 2004, 15: 445454

[18] Bulboaca T. A class of superordination-preserving integral operators. Indag Math N S, 2002, 13: 301311

[19] Ali R M, Ravichandran V, Seenivasagan N. Subordination and superordination of the Liu-Srivastava linear

operator on meromorphic functions. Bull Malays Math Sci Soc (2), 2008, 31(2): 192207

[20] Ali R M, Ravichandran V, Seenivasagan N. Subordination and superordination on Schwarzian derivatives.

J Inequal Appl, 2008, Art. ID 712328, 18 pp

[21] Ali R M, Ravichandran V, Seenivasagan N. Differential subordination and superordination of analyticfunctions defined by the multiplier transformation. Math Inequal Appl, 2009, 12: 123139

[22] Ali R M, Ravichandran V, Seenivasagan N. Differential subordination and superordination for meromorphic

functions defined by certain multiplier transformation. Bull Malays Math Sci Soc (2), 2010, 33(2): 311324

[23] Cho N E, Kwon O S, Owa S, Srivastava H M. A class of integral operators preserving subordination and

superordination for meromorphic functions. Appl Math Comput, 2007, 193: 463474

[24] Cho N E, Kwon O S. A class of integral operators preserving subordination and superordination. Bull

Malays Math Sci Soc (2), 2010, 33(3): 429437

[25] Wang Z G, Xiang R G, Darus M. A family of integral operators preserving subordination and superordi-

nation. Bull Malays Math Sci Soc (2), 2010, 33(1): 121131

[26] Miller S S, Mocanu P T. Differential subordinations and univalent functions. Michigan Math J, 1981, 28:

157171

[27] Miller S S, Mocanu P T. Univalent solutions of Briot-Bouquet differential equations. J Different Eq, 1985,

567: 297309

[28] Pommerenke C. Univalent Functions. Vanderhoeck and Ruprecht: Gottingen, 1975

[29] Kaplan W. Close-to-convex schlicht functions. Michigan Math J, 1952, 2: 169185

[30] Goel R M, Sohi N S. On a class of meromorphic functions. Glas Mat, 1982, 17(37): 1928

[31] Kumar V, Shukla S L. Certain integrals for classes of p-valent functions. Bull Austral Math Soc, 1982,

25: 8597

[32] Srivastava H M, Owa S. Current Topics in Analytic Function Theory. Singapore, New Jersey, London,

and Hong Kong: World Scientific Publishing Company, 1992

actamathscientia32b(3)(2012)929-941cho

Documents