- Research Article
- Open Access

# Univalence of Certain Linear Operators Defined by Hypergeometric Function

- R. Aghalary
^{1}and - A. Ebadian
^{1}Email author

**2009**:807943

https://doi.org/10.1155/2009/807943

© R. Aghalary and A. Ebadian. 2009

**Received:**11 January 2009**Accepted:**22 April 2009**Published:**2 June 2009

## Abstract

The main object of the present paper is to investigate univalence and starlikeness of certain integral operators, which are defined here by means of hypergeometric functions. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

## Keywords

- Positive Integer
- Analytic Function
- Integral Operator
- Unit Disk
- Fractional Derivative

## 1. Introduction and Preliminaries

where and

The special case of this class has been studied by Ponnusamy and Vasundhra [1] and Obradović et al. [2].

For a,b,c *∈* C and c*≠* 0,-1,-2,*…*, the Gussian hypergeometric series F(a,b;c;z) is defined as

This operator has been studied by Srivastava et al. [4] and Srivastava and Mishra [5].

This operator has been investigated by many authors such as Trimble [6], and Obradović et al. [7].

where and

where and

In this investigation we aim to find conditions on such that implies that the function to be starlike. Also we find conditions on for each the transforms and belong to and .

For proving our results we need the following lemmas.

Lemma 1.1 (cf. Hallenbeck and Ruscheweyh [8]).

and is the best dominant of (1.20).

Lemma 1.2 (cf. Ruscheweyh and Stankiewicz [8]).

If are analytic and are convex functions such that then .

Lemma 1.3 (cf. Ruscheweyh and Sheil-Small [9]).

Let and be univalent convex functions in . Then the Hadamard product is also univalent convex in .

## 2. Main Results

We follow the method of proof adopted in [1, 10].

Theorem 2.1.

Proof.

Suppose that denote the class of all Schwarz functions such that and let

Define by

Case 1.

Case 2.

Now the required conclusion follows from (2.13) and (2.14).

By putting in Theorem 2.1 we obtain the following result.

Corollary 2.2.

Let be the positive integer with . Also let and . If belongs to , then whenever .

Remark 2.3.

Taking in Theorem 2.1 and Corollary 2.2 we get results of [10].

We follow the method ofproofadopted in [11].

Theorem 2.4.

then the transform defined by (1.16) has the following:

(1)

Proof.

and the result follows from the last subordination and Corollary 2.2.

It is well-known that (see, [12]) if and , then is univalent convex function in . So if we take in the Theorem 2.4, we obtain the following.

Corollary 2.5.

Then the transform defined by (1.16) has the following:

(1)

By putting on the (1.8), we get which is evidently convex. So by taking on Theorem 2.4 we have the following.

Corollary 2.6.

Then the transform defined by (1.16) has the following:

(1) ;

Remark 2.7.

Taking and on Corollary 2.6, we get a result of [11].

By putting and on Theorem 2.10 we obtain the following.

Corollary 2.8.

then we have the following:

(1)

Remark 2.9.

We note that if , then is convex function, and so we can replace with in Corollary 2.8 to get other new results.

In [13], Pannusamy and Sahoo have also considered the class for the case with

Theorem 2.10.

Then the transform defined by (1.17) has the following:

(1)

Proof.

and the result follows from (2.46) and Corollary 2.2.

## Authors’ Affiliations

## References

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## Copyright

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