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# Some Applications of Srivastava-Attiya Operator to *p*-Valent Starlike Functions

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 790730 (2010)

## Abstract

We introduce and study some new subclasses of *p*-valent starlike, convex, close-to-convex, and quasi-convex functions defined by certain Srivastava-Attiya operator. Inclusion relations are established, and integral operator of functions in these subclasses is discussed.

## 1. Introduction

Let denote the class of functions of the form

which are analytic and *p*-valent in the open unit disc Also, let the Hadamard product or (convolution) of two functions

be given by

A function is said to be in the class of *p*-valent functions of order if it satisfies

we write the class of *p*-valent starlike in .

A function is said to be in the class of *p*-valent convex functions of order if it satisfies

The class of *p*-valent convex functions in is denoted by

It follows from (1.3) and (1.4) that

The classes and were introduced by Goodman [1]. Furthermore, a function is said to be *p*-valent close-to-convex of order and type in if there exists a function such that

We denote this class by The class was studied by Aouf [2]. We note that was studied by Libera [3].

A function is called quasi-convex of order type if there exists a function such that

where We denote this class by . Clearly The generalized Srivastava-Attiya operator in [4] is introduced by

where

It is not difficult to see from (1.8) and (1.9) that

When the operator is well-known Srivastava-Attiya operator [5].

Using the operator we now introduce the following classes:

In this paper, we will establish inclusion relation for these classes and investigate Srivastava-Attiya operator for these classes.

We note that

for we get Jung-Kim-Srivastava ([6, 7]);

for we get the generalized Libera integral operator. [8, 9];

for being any negative integer, and , the operator was studied by Slgean [10].

## 2. Inclusion Relation

In order to prove our main results, we will require the following lemmas.

Lemma 2.1 (see [11]).

Let be regular in U with . If attains its maximum value on the circle at a given point then where is a real number and

Lemma 2.2 (see [12]).

Let and let be a complex function, Suppose that satisfies the following conditions:

is continuous in D,

and

for al such that

Let be analytic in , such that for If then for

Our first inclusion theorem is stated as follows.

Theorem 2.3.

for any complex number .

Proof.

Let and set

where Using the identity

we have

Differentiating (2.3), logarithmically with respect to , we obtain

Now, from the function by taking in (2.4) as

it is easy to see that the function satisfies condition (i) and (ii) of Lemma 2.2, in . To verify condition (iii), we calculate as follows:

where and Therefore, the function satisfies the conditions of Lemma 2.2.

This shows that if then

if then

This completes the proof of Theorem 2.3.

Theorem 2.4.

, for any complex number s.

Proof.

Consider the following:

which evidently proves Theorem 2.4.

Theorem 2.5.

, for any complex number s.

Proof.

Let Then, there exists a function such that

Taking the function which satisfies we have and

Now, put where Using the identity (2.2) we have

Since and , we let , where thus (2.11) can be written as

Consider that

Differentiating both sides of (2.13), and multiplying by , we have

Using (2.14) and (2.12), we get

Taking in (2.15), we form the function as

It is not difficult to see that satisfies the conditions (i) and (ii) of Lemma 2.2 in To verify condition (iii), we proceed as follows:

where and being the functions of and and

By putting , we have

Hence, and The proof of Theorem 2.5 is complete

Theorem 2.6.

for any complex number s.

Proof.

Consider the following:

The proof of Theorem 2.6 is complete.

## 3. Integral Operator

For and , we recall here the generalized Bernardi-Libera-Livingston integral operator as follows

The operator when was studied by Bernardi [13], for was investigated earlier by Libera [14]. Now, we have

so we get the identity

The following theorems deal with the generalized Bernard-Libera-Livingston integral operator defined by (3.1).

Theorem 3.1.

Let If , then

Proof.

From (3.3), we have

where is analytic in , Using (3.3) and (3.4) we get

Differentiating (3.5), we obtain

Now we assume that Otherwise, there exists a point such that Then by Lemma 2.1, we have Putting and in (3.6), we have

which contradicts the hypothesis that

Hence, , for and it follows (3.4) that

The proof of Theorem 3.1 is complete·

Theorem 3.2.

Let If then

Proof.

Consider the following:

This completes the proof of Theorem 3.2.

Theorem 3.3.

Let If then

Proof.

Let Then, by definition, there exists a function such that

Then,

where From (3.3) and (3.10), we have

Since then from Theorem 3.1, we have

Let

where Using (3.11), we have

Also, (3.10) can be written as

Differentiating both sides, we have

or

Now, from (3.13) we have

We form the function by taking in (3.17) as follows

It is clear that the function defined in by (3.18) satisfies conditions (i) and (ii) of Lemma 2.2. To verify the condition(iii), we proceed as follows:

where and being the functions of and and

By putting , we have

Hence, and . Thus, we have The proof of Theorem 3.3 is complete.

Theorem 3.4.

Let If , then

Proof.

Consider the following:

and the proof of Theorem 3.4 is complete.

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## Acknowledgement

The authors would like to thank the referees of the paper for their helpful suggestions.

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Elrifai, E., Darwish, H. & Ahmed, A. Some Applications of Srivastava-Attiya Operator to *p*-Valent Starlike Functions.
*J Inequal Appl* **2010**, 790730 (2010). https://doi.org/10.1155/2010/790730

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DOI: https://doi.org/10.1155/2010/790730