Journal of Mathematics

Volume 2013 (2013), Article ID 630934, 4 pages

http://dx.doi.org/10.1155/2013/630934

Research Article

Coefficient Inequalities of Analytic Functions Related to Robertson Functions

Muhammad Arif and Mumtaz Ali

Department of Mathematics, Abdul Wali Khan University, 23200 Mardan, Khyber Pakhtunkhwa, Pakistan

Received 5 January 2013; Revised 19 March 2013; Accepted 2 April 2013

Academic Editor: S. T. Ali

Copyright © 2013 Muhammad Arif and Mumtaz Ali. 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.

We introduce and study a subclass of analytic functions related to Robertson functions. Here we discuss the coefficient estimate for function in this class.

1. Introduction

Let be the class of functions of the form

which are analytic in the open unit disc . Also let and denote the well-known classes of starlike and convex functions, respectively.

For any two analytic functions given by (1) and with

the convolution (Hadamard product) is given by

Using the concept of convolution, Ruscheweyh [1] introduced a differential operator given by

with

where is a Pochhammer symbol given as

It is obvious that , , and

The following identity can easily be established:

Now with the help of Ruscheweyh derivative, we define a class of analytic functions as follows.

Definition 1. Let . Then, , if and only if

where , , , is real with , and .

By giving specific values to , , , , and in , we obtain many important subclasses studied by various authors in earlier papers, see for details [2–5], and list some of them as follows:

(i) and , studied by Spacek [6] and Robertson [7], respectively; for the advancement work, see [8, 9].(ii) and , studied by both Owa et al. and Shams et al. [10, 11].(iii), , introduced by Ravichandran et al. [12]. (iv), considered by Latha [13].(v), , the well-known classes of starlike and convex functions of order .

From the above special cases, we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.

We will assume throughout our discussion, unless otherwise stated, that , , , is real with , and .

2. Some Properties of the Class

Theorem 2. If with , then

Proof. Since for any complex number , implies that

which implies that

And hence, we obtain the required result.

Put , , and in Theorem 2; we obtain the following result.

Corollary 3 (see [10]). If with , then

Set , , and in Theorem 2; one has the following result.

Corollary 4 (see [10]). If with , then

Theorem 5. If , then

where

Proof. We note that for ,

Let us define the function by

Then, is analytic in with and . Let

Then, (19) can be written as

and using (8), we have

which implies that

where we have used (4) and (5). Now applying the coefficient estimates for Caratheodory function [14], we obtain

For ,

which proves (15).

For ,

Therefore, (16) holds for . Assume that (16) is true for all and consider

Thus, the result is true for , and hence by induction, (16) holds for all .

If we set , , and in Theorem 5, we get the result proved in [10].

Corollary 6. If , then

Remark 7. If we take in Corollary 6, we have

which was proved by Robertson [15].

By setting , , and in Theorem 5, we obtain the result in [10].

Corollary 8. If , then

Remark 9. Letting in Corollary 8, we have

given by Robertson [15].

Acknowledgment

The principle author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.

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