- Research Article
- Open Access
- Published:

# Univalence of Certain Linear Operators Defined by Hypergeometric Function

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 807943 (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

Let denote the class of all analytic functions in the unit disk . For , a positive integer, let

with , where is referred to as the normalized analytic functions in the unit disc. A function is called starlike in if is starlike with respect to the origin. The class of all starlike functions is denoted by . For , we define

and it is called the class of all starlike functions of order . Clearly, for . For functions , given by

we define the Hadamard product (or convolution) of and by

An interesting subclass of (the class of all analytic univalent functions) is denoted by and is defined by

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

where and . It is well-known that is analytic in . As a special case of the Euler integral representation for the hypergeometric function, we have

Now by letting

it is easily seen that

For , Owa and Srivastava [3] introduced the operator defined by

which is extensions involving fractional derivatives and fractional integrals. Using definition of we may write

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

Also for and , let us define the function by

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

If we take

then we can rewrite operator defined by (1.11) as

From the definition of it is easy to check that

For with for all we define the transform by

where and

Also for with for all we define the transform by

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]).

Let be analytic and convex univalent in the unit disk with . Also let

be analytic in . If

then

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.

Let n be positive integer with . Also let and . If belongs to , Then whenever , where

Proof.

Let us define

Since , we have

where is an analytic function with and By Schwarz lemma, we have . By (2.3), it is easy to check that

Therefore

We need to show that . To do this, according to a well-known result [9] and (2.5) it suffices to show that

which is equivalent to

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

then, if . This observation shows that it suffices to find . First we notice that

Define by

Differentiating with respect to , weget

Case 1.

Let Then we see that has its only critical point in the positive real line at

Furthermore, we can see that for and for . Hence attains its maximum value at and

Case 2.

Let , then it is easy to see that and so attains its maximum value at and

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.

Let with and the function with be univalent convex in . If and defined by (1.8) satisfy the conditions

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

(1)

(2)*whenever*

Proof.

From the definition of we obtain

Differentiating shows that

It is easy to see that

From (1.9) and (2.19) we deduce that

or

Let us define

then is analytic in , with and Combining (2.18) with (2.21), one can obtain

Differentiating yields

In view of (2.21), (2.23), and (2.24), we obtain

Hence

Since and are convex and

by using Lemmas 1.2 and 1.3, from (2.26) we deduce that

It now follows from Lemma 1.1 that

Therefore

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.

For and , let the function and defined by (1.8) satisfy the condition

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

(1)

(2) whenever

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.

For with , let the function and defined by (1.8) satisfy the condition

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

(1);

(2)*whenever*

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.

Let and with be univalent convex function in . Also let with and , satisfy

and let be the function which is defined by

If

then we have the following:

(1)

(2) whenever

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.

For let and defined by (1.13) satisfy the condition

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

(1)

(2) whenever

Proof.

Let us define

then is analytic in , with and Using the same method as on Theorem 2.4 we get

Since and are convex,

Using Lemmas 1.2 and 1.3, from (2.42) it yields

It now follows from Lemma 1.1 that

Therefore

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

## References

Ponnusamy S, Vasundhra P:

**Criteria for univalence, starlikeness and convexity.***Annales Polonici Mathematici*2005,**85**(2):121–133. 10.4064/ap85-2-2Obradović M, Ponnusamy S, Singh V, Vasundhra P:

**Univalency, starlikeness and convexity applied to certain classes of rational functions.***Analysis*2002,**22**(3):225–242.Owa S, Srivastava HM:

**Univalent and starlike generalized hypergeometric functions.***Canadian Journal of Mathematics*1987,**39**(5):1057–1077. 10.4153/CJM-1987-054-3Srivastava HM, Mishra AK, Das MK:

**A nested class of analytic functions defined by fractional calculus.***Communications in Applied Analysis*1998,**2**(3):321–332.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)00333-8Trimble SY:

**The convex sum of convex functions.***Mathematische Zeitschrift*1969,**109:**112–114. 10.1007/BF01111242Obradović M, Ponnusamy S, Vasundhra P:

**Univalence, strong starlikeness and integral transforms.***Annales Polonici Mathematici*2005,**86**(1):1–13. 10.4064/ap86-1-1Ruscheweyh St, Stankiewicz J:

**Subordination under convex univalent functions.***Bulletin of the Polish Academy of Sciences, Mathematics*1985,**33**(9–10):499–502.Ruscheweyh St, Sheil-Small T:

**Hadamard products of Schlicht functions and the Pölya-Schoenberg conjecture.***Commentarii Mathematici Helvetici*1973,**48:**119–135. 10.1007/BF02566116Ponnusamy S, Sahoo P:

**Geometric properties of certain linear integral transforms.***Bulletin of the Belgian Mathematical Society. Simon Stevin*2005,**12**(1):95–108.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.2007.03.020Ling Y, Liu F, Bao G:

**Some properties of an integral transform.***Applied Mathematics Letters*2006,**19**(8):830–833. 10.1016/j.aml.2005.10.012Ponnusamy 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Aghalary, R., Ebadian, A. Univalence of Certain Linear Operators Defined by Hypergeometric Function.
*J Inequal Appl* **2009**, 807943 (2009). https://doi.org/10.1155/2009/807943

Received:

Accepted:

Published:

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

### Keywords

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