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Journal of Complex Analysis

Volume 2014 (2014), Article ID 984135, 3 pages

http://dx.doi.org/10.1155/2014/984135

## An Application of a Poisson Distribution Series on Certain Analytic Functions

Department of Mathematics, U.I.E.T., C.S.J.M. University, Kanpur, Uttar Pradesh 208024, India

Received 25 November 2013; Accepted 7 January 2014; Published 18 February 2014

Academic Editor: Janne Heittokangas

Copyright © 2014 Saurabh Porwal. 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

The purpose of the present paper is to introduce a Poisson distribution series and obtain necessary and sufficient conditions for this series belonging to the classes and . We also consider an integral operator related to this series.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk and and satisfy the normalization condition . Further, we denote by the subclass of consisting of functions of the form (1) which are also univalent in and let be the subclass of consisting of functions of the form Let be the subclass of consisting of functions which satisfy the condition for some (), () and for all .

Also, we let denote the subclass of consisting of functions which satisfy the condition for some (), () and for all .

From (3) and (4) it is easy to verify that The classes and were extensively studied by Altintas and Owa [1] and certain conditions for hypergeometric functions and generalized Bessel functions for these classes were studied by Mostafa [2] and Porwal and Dixit [3].

It is worthy to note that , the class of starlike functions of order () and , the class of convex functions of order () (see [4]).

A variable is said to have Poisson distribution if it takes the values with probabilities , , , , respectively, where is called the parameter.

Thus Now, we introduce a power series whose coefficients are probabilities of the Poisson distribution: We note that, by ratio test, the radius of convergence of the above series is infinity.

Now, we introduce the series

Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [5–10]) and generalized Bessel functions (see [3, 11–13]), we obtain necessary and sufficient conditions for function belonging to the classes and . Finally, we give conditions for an integral operator belonging to the classes and .

#### 2. Main Results

To establish our main results, we will require the following Lemmas according to Altintas and Owa [1].

Lemma 1 (see [1]). *A function defined by (2) is in the class if and only if
*

Lemma 2 (see [1]). *A function defined by (2) is in the class if and only if
*

Theorem 3. *If , then is in , if and only if
*

*Proof. *Since
according to Lemma 1, we must show that

Now
But this last expression is bounded previously by if and only if (11) holds.

Thus the proof of Theorem 3 is established.

Theorem 4. *If , then is in , if and only if
*

*Proof. *Since
according to Lemma 2, we must show that
Now
But this last expression is bounded above by if and only if (15) holds. This completes the proof of Theorem 4.

#### 3. An Integral Operator

In the following theorem, we obtain similar results in connection with a particular integral operator as follows:

Theorem 5. *If , then defined by (19) is in if and only if
*

*Proof. *Since
by Lemma 2, we need only to show that
Now
which is bounded above by , if and only if (20) holds.

Theorem 6. *If , then defined by (19) is in if and only if
*

*Proof. *The proof of this theorem is similar to that of Theorem 5. Therefore we omit the details involved.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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