Abstract and Applied Analysis

Volume 2012, Article ID 259054, 10 pages

http://dx.doi.org/10.1155/2012/259054

## On an Integral Transform of a Class of Analytic Functions

Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, India

Received 5 June 2012; Accepted 14 September 2012

Academic Editor: Josip E. Pecaric

Copyright © 2012 Sarika Verma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For and , let denote the class of all normalized analytic functions in the open unit disc such that , for some . It is known (Noshiro (1934) and Warschawski (1935)) that functions in are close-to-convex and hence univalent for . For , we consider the integral transform , where is a nonnegative real-valued integrable function satisfying the condition . The aim of present paper is, for given , to find sharp values of such that (i) whenever and (ii) whenever .

#### 1. Introduction

Let denote the class of analytic functions defined in the open unit disc with the normalizations , and let be the subclass of consisting of functions univalent in . For any two functions and in , the Hadamard product (or convolution) of and is the function defined by For , Fournier and Ruscheweyh [1] introduced the integral operator where is a nonnegative real-valued integrable function satisfying the condition . This operator contains some well-known operators such as Libera, Bernardi, and Komatu as its special cases. Fournier and Ruscheweyh [1] applied the famous duality theory to show that for a function in the class the linear integral operator is univalent in . Since then, this operator has been studied by a number of authors for various choices of . In another remarkable paper, Barnard et al. in [2] obtained conditions such that whenever is in the class with . Note that for , functions in satisfy the condition in and thus are close-to-convex in . A domain in is close-to-convex if its compliment in can be written as union of nonintersecting half lines.

In 2008, Ponnusamy and Rønning [3] discussed the univalence of for the functions in the class In a very recent paper, Ali et al. [4] studied the class where and . In this paper, they obtained sufficient conditions so that the integral transform maps normalized analytic functions into the class of starlike functions. It is evident that , and .

In the present paper, we shall mainly tackle the following problems. (1)For given , find sharp values of such that whenever . (2)For given , find sharp values of such that whenever . To prove one of our results, we shall need the generalized hypergeometric function , so we define it here.

Let and be complex numbers with . Then the generalized hypergeometric function is defined by where is the Pochhammer symbol, defined in terms of the Gamma function, by In particular, is called the Gaussian hypergeometric function. We note that the series in (1.7) converges absolutely for if and for if .

We shall also need the following lemma.

Lemma 1.1 (see [5]). * Let , , and . Then, for analytic in with , the conditions and imply , where .*

#### 2. Main Results

We use the notations introduced in [4]. Let and satisfy When , then is chosen to be 0, in which case, . When , (2.1) yields or .(i)For , then choosing gives .(ii)For , then and .

Theorem 2.1. *Let , satisfy (2.1). Further, let be given, and define by
**
If , then . The value of is sharp.*

* Proof. * The case corresponds to Theorem 1.5 in [2]. So we assume that .

Define
Writing , it follows that
It is a simple exercise to see that
Let , where is defined by (1.2). Then for , we can write

Since , it follows that for some . Now, for each , we first claim that
which, by Lemma 1.1, implies that . Therefore, it suffices to verify the inequality (2.7). Using the identity (which can be checked by comparing the coefficients of on both sides)
it follows that
Thus,
Therefore, for , we have
in the view of .

To prove the sharpness, let be the function determined by
Using a series expansion, we see that we can write
Then,
where . Equation can be restated as
Finally,
which for takes the value
This shows that the result is sharp.

Letting and in Theorem 1.1, we obtain the following result of Ruscheweyh [6].

Corollary 2.2. * Let , and define by
**
If , then . The value of is sharp.*

Theorem 2.3. * Let and , and define by
**
If , then . The value of is sharp.*

* Proof. * The idea of the proof is similar to the one used to prove Theorem 2 in [1].

Let . Clearly,
Since, , so with
we have , where .

For ,
Putting this value in (2.20),
Equivalently,
Thus
In the case when ,
Since
This leads to,
which is clearly (2.25) with .

Further if and only if . Now using (2.25), we obtain
Since for some , it follows by duality principle [8, page 23] that
if, and only if,
Using , we get
By using (2.19), we have
Thus,
which implies that
Thus, we deduce, using duality principle, that is contained in a half plane not containing the origin. So, and hence .

To prove the sharpness, let .
Further,
gives
or
Further, assume that
Since ,

so,
Therefore, for ,
This shows that the result is sharp.

Letting in Theorem 2.3 above, we obtain the following result of Kim and Rønning [9].

Corollary 2.4. * Let and , and define by
**
If , then . The value of is sharp.**Upon setting with , we have the following corollary.*

Corollary 2.5. * Let , , and be given, and let be defined by
**
Suppose that , then , where
**
The constant is sharp.*

The special case of Corollary 2.5 (with ) has been obtained by Aghalary et al. [11].

#### References

- R. Fournier and S. Ruscheweyh, “On two extremal problems related to univalent functions,”
*The Rocky Mountain Journal of Mathematics*, vol. 24, no. 2, pp. 529–538, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. W. Barnard, S. Naik, and S. Ponnusamy, “Univalency of weighted integral transforms of certain functions,”
*Journal of Computational and Applied Mathematics*, vol. 193, no. 2, pp. 638–651, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ponnusamy and F. Rønning, “Integral transforms of a class of analytic functions,”
*Complex Variables and Elliptic Equations*, vol. 53, no. 5, pp. 423–434, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Ali, A. O. Badghaish, V. Ravichandran, and A. Swaminathan, “Starlikeness of integral transforms and duality,”
*Journal of Mathematical Analysis and Applications*, vol. 385, no. 2, pp. 808–822, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ponnusamy, “Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane,”
*The Rocky Mountain Journal of Mathematics*, vol. 28, no. 2, pp. 695–733, 1998, Reports of the Department of Mathematics, Preprint 92 , University of Helsinki, Finland, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ruscheweyh, “Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc,”
*Transactions of the American Mathematical Society*, vol. 210, pp. 63–74, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. E. Warschawski, “On the higher derivatives at the boundary in conformal mapping,”
*Transactions of the American Mathematical Society*, vol. 38, no. 2, pp. 310–340, 1935. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ruscheweyh,
*Convolutions in Geometric Function Theory*, vol. 83 of*Séminaire de Mathématiques Supérieures*, Presses de l'Université de Montréal, Montreal, Canada, 1982. - Y. C. Kim and F. Rønning, “Integral transforms of certain subclasses of analytic functions,”
*Journal of Mathematical Analysis and Applications*, vol. 258, no. 2, pp. 466–489, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Noshiro, “On the theory of schlicht functions,”
*Journal of the faculty of science, Hokkaido University*, vol. 2, pp. 129–155, 1934. View at Google Scholar - R. Aghalary, A. Ebadian, and S. Shams, “Geometric properties of some linear operators defined by convolution,”
*Tamkang Journal of Mathematics*, vol. 39, no. 4, pp. 325–334, 2008. View at Google Scholar · View at Zentralblatt MATH