# 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.

## 1. Introduction and Preliminaries

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].

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

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:

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:

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:

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:

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:

Proof.

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

## Authors’ Affiliations

## References

- Ponnusamy S, Vasundhra P:
**Criteria for univalence, starlikeness and convexity.***Annales Polonici Mathematici*2005,**85**(2):121–133. 10.4064/ap85-2-2MathSciNetView ArticleMATHGoogle Scholar - Obradović M, Ponnusamy S, Singh V, Vasundhra P:
**Univalency, starlikeness and convexity applied to certain classes of rational functions.***Analysis*2002,**22**(3):225–242.MathSciNetMATHGoogle Scholar - Owa S, Srivastava HM:
**Univalent and starlike generalized hypergeometric functions.***Canadian Journal of Mathematics*1987,**39**(5):1057–1077. 10.4153/CJM-1987-054-3MathSciNetView ArticleMATHGoogle Scholar - Srivastava HM, Mishra AK, Das MK:
**A nested class of analytic functions defined by fractional calculus.***Communications in Applied Analysis*1998,**2**(3):321–332.MathSciNetMATHGoogle Scholar - Srivastava HM, Mishra AK:
**Applications of fractional calculus to parabolic starlike and uniformly convex functions.***Computers & Mathematics with Applications*2000,**39**(3–4):57–69. 10.1016/S0898-1221(99)00333-8MathSciNetView ArticleMATHGoogle Scholar - Trimble SY:
**The convex sum of convex functions.***Mathematische Zeitschrift*1969,**109:**112–114. 10.1007/BF01111242MathSciNetView ArticleMATHGoogle Scholar - Obradović M, Ponnusamy S, Vasundhra P:
**Univalence, strong starlikeness and integral transforms.***Annales Polonici Mathematici*2005,**86**(1):1–13. 10.4064/ap86-1-1MathSciNetView ArticleMATHGoogle Scholar - Ruscheweyh St, Stankiewicz J:
**Subordination under convex univalent functions.***Bulletin of the Polish Academy of Sciences, Mathematics*1985,**33**(9–10):499–502.MathSciNetMATHGoogle Scholar - Ruscheweyh St, Sheil-Small T:
**Hadamard products of Schlicht functions and the Pölya-Schoenberg conjecture.***Commentarii Mathematici Helvetici*1973,**48:**119–135. 10.1007/BF02566116MathSciNetView ArticleMATHGoogle Scholar - Ponnusamy S, Sahoo P:
**Geometric properties of certain linear integral transforms.***Bulletin of the Belgian Mathematical Society. Simon Stevin*2005,**12**(1):95–108.MathSciNetMATHGoogle Scholar - Obradović M, Ponnusamy S:
**Univalence and starlikeness of certain transforms defined by convolution of analytic functions.***Journal of Mathematical Analysis and Applications*2007,**336**(2):758–767. 10.1016/j.jmaa.2007.03.020MathSciNetView ArticleMATHGoogle Scholar - Ling Y, Liu F, Bao G:
**Some properties of an integral transform.***Applied Mathematics Letters*2006,**19**(8):830–833. 10.1016/j.aml.2005.10.012MathSciNetView ArticleMATHGoogle Scholar - Ponnusamy S, Sahoo P:
**Special classes of univalent functions with missing coefficients and integral transforms.***Bulletin of the Malaysian Mathematical Sciences Society*2005,**28**(2):141–156.MathSciNetMATHGoogle Scholar

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