Journal of Mathematics

Volume 2013 (2013), Article ID 784560, 6 pages

http://dx.doi.org/10.1155/2013/784560

Research Article

Certain Properties of Multivalent Functions Associated with the Dziok-Srivastava Operator

T. M. Seoudy

Department of Mathematics, Faculty of Science, Fayoum 63514, Egypt

Received 11 October 2012; Accepted 26 October 2012

Academic Editor: Harold Benson

Copyright © 2013 T. M. Seoudy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By making use of the techniques of the differential subordination, we derive certain properties of -valent functions associated with the Dziok-Srivastava operator.

1. Introduction

Let denote the class of functions of the form

which are analytic in the open unit disk  . We write .

Suppose that and are analytic in . We say that the function is subordinate to in , or superordinate to in , and we write or , if there exists an analytic function in with and , such that . If is univalent in , then the following equivalence relationship holds true (see [1–3]):

For functions given by

we define the Hadamard product (or convolution) of and by

For complex parameters and , the generalized hypergeometric function is defined (see [4]) by the following infinite series:

where is the Pochhammer symbol defined, in terms of the Gamma function , by

Corresponding a function defined by

Dziok and Srivastava [5] considered a linear operator

defined by the following Hadamard product:

If is given by (1), then we have

where

To make the notation simple, we write

It easily follows from (9) or (10) that

It should be remarked that the linear operator is a generalization of many other linear operators considered earlier. In particular, for we have the following observations:

(i), where the linear operator was investigated by Hohlov [6];(ii), where the linear operator was studied by Goel and Sohi [7]. In the case when , is the Ruscheweyh derivative of (see [8]);(iii), where is the generalized Bernardi-Libera-Livingston integral operator (see [9]);(iv), where is the fractional integral of of order when and fractional derivative of of order when . The extended fractional differintegral operator was introduced and studied by Patel and Mishra [10]. The fractional differential operator with was investigated by Srivastava and Aouf [11]. The operator was introduced by Owa and Srivastava [12] (see also [13–15]).(v) = , where the linear operator was studied by Saitoh [16] which yields the operator introduced by Carlson and Shaffer [17] for ;(vi) = , where is the Choi-Saigo-Srivastava operator [9] which is closely related to the Carlson-Shaffer [17] operator ;(vii) = , where the operator was considered by Liu and Noor [18];(viii) = , where is the Cho-Kwon-Srivastava operator [19].

In recent years, many interesting subclasses of analytic functions, associated with the Dziok-Srivastava operator and its many special cases, were investigated by, for example, Dziok and Srivastava [5, 20], Gangadharan et al. [21], Liu and Noor [18], Liu [22], Liu and Srivastava [23], and others see also [19, 24–26]). In the present paper, we shall use the method based upon the differential subordination to derive inclusion relationships and other interesting properties and characteristics of the Dziok-Srivastava operator .

2. Main Results

Unless otherwise mentioned, we assume throughout the sequel that ; ; ; and .

Let denote the class of functions of the form

that are analytic in , we write . In our present investigation, we shall require the following lemmas.

Lemma 1 (see [2]). Let be analytic and convex (univalent) in with and . If

then, for and ,

and is the best dominant.

Lemma 2 (see [1]). Let be a set in the complex plane and be a complex number satisfying . Suppose that the function satisfies the condition for all real and for all . If the functions and , then in .

Lemma 3 (see [27]). Let be analytic in with and for all . If there exist two points such that

for some and and for all , then

where

Theorem 4. Let , . Let , then

implies

The bound is the best possible.

Proof. It easily follows from (13) that

From (20) and (22), we have

That is,

Let

then (24) may be written as

By using a well-known result (see [28]) to (26) we obtain that

or, equivalently,

where is analytic in , and for . Since for and , (28) yields

To see that the bound cannot be increased, we consider the function

Since

we easily have that satisfies (20) and

as . This completes the proof of Theorem 4.

Theorem 5. Let , . If satisfies the following inequality

then

where is the positive root of the equation

Proof. Let

then is analytic in and . Differentiating (36) and using (22), we obtain that

where

Using (33) and (38), we have

Now for all real , we have

where is the positive root of (35).

Note that for , , and

we have and . This shows . Hence for each , . By Lemma 2, we get , and this proves (34).

Theorem 6. Suppose that ; and , . If given by

satisfies

then

where and are the solution of the equations:

where is given by (19).

Proof. Using (42) and the identity (22), it follows that

for . Putting

On differentiating (47) followed by a simple calculation, we get

Let be the function which maps onto the angular domain with . By using (43) in (48), we get

Further, an application of Lemma 1 yields in and hence for .

Suppose there exist two points such that the condition (28) is satisfied. Then by Lemma 3, we obtain (18) under the constraint (19). Therefore, we have

which contradicts the assumption (43). This proves the assertion (44) of the Theorem 6.

For , Theorem 6 reduces to the following corollary.

Corollary 7. Suppose that and . If defined by (42) satisfies

then

where is the solution of the equation:

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