Chinese Journal of Mathematics

Volume 2014 (2014), Article ID 974175, 7 pages

http://dx.doi.org/10.1155/2014/974175

Research Article

Inclusion Properties of New Classes of Analytic Functions

Mohan Das, Ramesha Chendady, and Indira Kasarkod Pal

Department of Mathematics, Manipal Institute of Technology, Manipal, Karnataka 576 104, India

Received 17 October 2013; Accepted 24 November 2013; Published 31 March 2014

Academic Editors: S. Deng and Y. Feng

Copyright © 2014 Mohan Das et al. 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.

The purpose of the present paper is to introduce certain new subclasses of analytic functions defined by Srivastava-Attiya operator and study their inclusion relationships and to obtain some interesting consequences of the inclusion relations.

1. Introduction, Definitions, and Preliminaries

Let be the class of analytic functions defined on the unit disk , normalized by the conditions . Let be the subclass of consisting of univalent functions. Let denote the class of analytic convex univalent functions defined in satisfying the conditions and for all .

For any two analytic functions and defined in the unit disc , we say that is subordinate to in , denoted by , if is univalent in , and .

The convolution or the Hadamard product of two analytic functions with power series representations

convergent in is the function , with power series representation

The following classes of functions were defined by Shanmugam [1].

For a fixed analytic function and ,

We note that .

When and , , the class of starlike functions; , the class of convex functions; and , the class of close to convex functions.

The generalized Hurwitz-Lerch Zeta function is defined as

where   .

This function contains, as its special cases, functions such as the Riemann and Hurwitz zeta function, Lerch zeta function, the polylogarithmic function, and the Lipschitz Lerch zeta function.

Several interesting properties and characteristics of the Hurwitz-Lerch zeta function can be found in the recent investigations by Choi and Srivastava [2], Ferreira and López [3], Garg et al. [4], Lin and Srivastava [5], and Lin et al. [6].

Using this function, Srivastava and Attiya [7] introduced the following family of linear operator , defined by

where the function is given by

Using (6) in (5), we get

For and ,

where denotes Alexander transform [8] and denotes Bernardi integral operator [9].

By (7), Srivastava and Attiya obtained the following relation:

In order to obtain our main results, we need the following lemmas.

Lemma 1 (see [10]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Then

Lemma 2 (see [11]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Let be analytic in with . Then

Lemma 3 (see [12]). Let a function be analytic in and . If there exists a point such that and , then

where

and .

2. Main Results

First we will define a subclass as follows.

Definition 4. For a fixed analytic function and a real ,

Remark 5. When , , and , reduces to the class .

Theorem 6. For ,   .

Proof. Let and .

Note that is analytic and .

Then,

Hence

where

Since .

Thus, by using Lemma 2,

That is, .

In the following, we obtain an inclusion relation of the class .

Theorem 7. For .

Proof. Let .

Then

and by Theorem 6,

Thus,

Since is a convex domain,

where .

That is,

Let . Then and (23) implies that .

Theorem 8. Let . If then , for .

Proof. Let .

We prove the result by mathematical induction on .

When , .

When , set

Clearly, is analytic and .

From (9), we get

By logarithmic differentiation and multiplication by , we get

As , by Lemma 1, we get .

That is, .

Suppose that .

Set

Clearly, is analytic and .

From (9), we get

By logarithmic differentiation and multiplication by , we get

Hence, by Lemma 1, , for .

That is, , which proves the result.

Theorem 9. Let . If then , for .

Proof. Let .

Then

Remark 10. The Bernardi integral operator preserves both the classes and whenever .

Corollary 11. Let , , and .

(i)If then .(ii)If then .

Theorem 12. Let and

Then

Proof. Let .

From (9), we get

From Theorem 8, for all.

Hence,

Let

Then for ,

This completes the proof.

In the following, we introduce certain classes of analytic functions defined by Srivastava-Attiya operator and investigate their inclusion relationships.

Let and .

Definition 13. For and , let

Definition 14. For and , let

Remark 15. Clearly, if and only if .

Definition 16. For and , let

Remark 17. We note that

Theorem 18. If , then .

Proof. Let . Then,

Let

where is analytic and .

Using (42) in (9), we get

On logarithmic differentiation and multiplication by , we arrive at

Since is convex univalent in with and , Lemma 1 and the subordination relation (44) gives

proving the theorem.

Theorem 19. If , then .

Proof. Let .

Then

Theorem 20. If , then .

Proof. Let . Then, by Definition 16, there exists a function such that

Using Theorem 18, we get .

Thus,

where is analytic.

Let

where is analytic and .

Now, making use of (9),

Simplifying the above, using (48) and (49), and then using (47) we arrive at

Since is convex univalent in with and , Lemma 2 and the subordination relation (51) gives

proving the assertion.

Now, we consider certain interesting consequences of the above inclusion relationships.

Corollary 21. Suppose that .

(i)If then .(ii)If then .(iii)If then .

Corollary 22. Suppose that and .

(i)If then .(ii)If then .(iii)If then .

In a similar manner one can obtain many applications of Theorems 18, 19, and 20.

Using Srivastava-Attiya integral operator, we introduce some classes of analytic functions, analogous to the class of strongly starlike functions of order and the class of strongly convex functions of order , and study the properties of the newly defined classes.

Let , , , , and .

Definition 23. Let

Definition 24. Let

Remark 25. Clearly, if and only if .

Remark 26. For , , is the class of strongly starlike functions of order and is the class of strongly convex functions of order .

Now we prove inclusion relationships for the above defined classes.

Theorem 27. Suppose that . Then for each .

Proof. Let .

We set

where is analytic in , and for all .

Using (9) and (55), we have

On logarithmic differentiation and multiplication by , we get

Suppose that there exists a point such that

Then, applying Lemma 3, we can write

Thus, if , then from (57)

This implies that

which contradicts the condition that .

Similarly, if arg , then we obtain that

which also contradicts the condition that .

Thus we have

That is,

Theorem 28. Suppose that . Then , for .

Proof. Let .

Then

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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