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

- EA Elrifai
^{1}, - HE Darwish
^{1}Email author and - AR Ahmed
^{1}

**2010**:790730

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

© E. A. Elrifai et al. 2010

**Received: **25 March 2010

**Accepted: **14 July 2010

**Published: **1 August 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.

## Keywords

## 1. Introduction

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

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

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

*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].

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

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 S l gean [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:

Let be analytic in , such that for If then for

Our first inclusion theorem is stated as follows.

Theorem 2.3.

Proof.

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

This completes the proof of Theorem 2.3.

Theorem 2.4.

Proof.

which evidently proves Theorem 2.4.

Theorem 2.5.

Proof.

Taking the function which satisfies we have and

where and being the functions of and and

Hence, and The proof of Theorem 2.5 is complete

Theorem 2.6.

Proof.

The proof of Theorem 2.6 is complete.

## 3. Integral Operator

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

Theorem 3.1.

Proof.

which contradicts the hypothesis that

Hence, , for and it follows (3.4) that

The proof of Theorem 3.1 is complete·

Theorem 3.2.

Proof.

This completes the proof of Theorem 3.2.

Theorem 3.3.

Proof.

Since then from Theorem 3.1, we have

where and being the functions of and and

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

Theorem 3.4.

Proof.

and the proof of Theorem 3.4 is complete.

## Declarations

### Acknowledgement

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

## Authors’ Affiliations

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