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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 415319, 7 pages

http://dx.doi.org/10.1155/2013/415319

## Coefficient Estimates and Other Properties for a Class of Spirallike Functions Associated with a Differential Operator

^{1}Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey^{2}Department of Mathematics, Faculty of Mathematics and Computer Science, Transylvania University of Braşov, Iuliu Maniu 50, 50091 Braşov, Romania^{3}Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, 34210 Istanbul, Turkey

Received 17 February 2013; Accepted 16 May 2013

Academic Editor: Adem Kılıçman

Copyright © 2013 Halit Orhan 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 , a new class of analytic functions defined by means of the differential operator is introduced. Our main object is to provide sharp upper bounds for Fekete-Szegö problem in . We also find sufficient conditions for a function to be in this class. Some interesting consequences of our results are pointed out.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk .

Let denote the subclass of consisting of functions that are univalent in .

A function is said to be in the class of -spirallike functions of order in , denoted by , if for and some real with .

The class was studied by Libera [1] and Keogh and Merkes [2].

Note that is the class of spirallike functions introduced by Špaček [3], is the class of starlike functions of order , and is the familiar class of starlike functions.

For the constants with and , denote The function maps the open unit disk onto the half-plane . If then it is easy to check that

For given by (1) and given by the Hadamard product (or convolution), denoted by , is defined by

Denote by the family of all analytic functions that satisfy the conditions and , .

A function is said to be subordinate to a function , written , if there exists a function such that , .

A classical theorem of Fekete and Szegö (see [4]) states that if is given by (1), then This inequality is sharp in the sense that for each there exists a function in such that the equality holds. Later Pfluger (see [5]) has considered the same problem but for complex values of . The problem of finding sharp upper bounds for the functional for different subclasses of is known as the Fekete-Szegö problem. Over the years, this problem has been investigated by many authors including [6–12].

For a function , we consider the following differential operator introduced by Răducanu and Orhan [13]:(i), (ii), (iii), where and .

If the function is given by (1), then, from the definition of the operator , it is easy to observe that where

It should be remarked that the operator generalizes other differential operators considered earlier. For , we have(i), the operator introduced by Sălăgean [14];(ii), the operator studied by Al-Oboudi [15].

In view of (9), can be written in terms of convolution as where

Define the function such that

It is easy to observe that

Making use of the differential operator , we define the following class of functions.

*Definition 1. * For , , and , denote by the class of functions which satisfy the condition

The class contains as particular cases the following classes of functions:
Also, the class consists of functions satisfying the inequality
An analogous of the class has been recently studied by Murugusundaramoorthy [16].

The main object of this paper is to obtain sharp upper bounds for the Fekete-Szegö problem for the class . We also find sufficient conditions for a function to be in this class.

#### 2. Membership Characterizations

In this section, we obtain several sufficient conditions for a function to be in the class .

Theorem 2. * Let , and let be a real number with . If
**
then provided that
*

*Proof. * From (18), it follows that
where . We have
provided that . Thus, the proof is completed.

If in Theorem 2 we take , we will obtain the following result.

Corollary 3. * Let . If
**
then . *

A sufficient condition for a function to be in the class , in terms of coefficients inequality, is obtained in the next theorem.

Theorem 4. *If a function given by (1) satisfies the inequality
**
where , , , and is defined by (10), then it belongs to the class . *

*Proof. * In virtue of Corollary 3, it suffices to show that the condition (22) is satisfied. We have

The last expression is bounded previously by , if
which is equivalent to

For special values of , , , and , from Theorem 4, we can derive the following sufficient conditions for a function to be in the classes , and , respectively.

Corollary 5. *Let . If
**
where , , and , then . *

Corollary 6 (see [17]). *Let . If
**
where , , then . *

Corollary 7 (see [18]). * Let . If
**
where , then . *

A necessary and sufficient condition for a function to be in the class can be given in terms of integral representation.

Theorem 8. *A function is in the class if and only if there exists such that
**
where and are defined by (3) and (13), respectively. *

*Proof. * In virtue of (15), if and only if there exists such that
From the last equality, we obtain
Making use of (14) and (32), we have
and thus, the proof is completed.

For , , define the function where and are defined by (3) and (13), respectively.

In virtue of Theorem 8, the function belongs to the class . Note that is an odd function.

#### 3. The Fekete-Szegö Problem

In order to obtain sharp upper bounds for the Fekete-Szegö functional for the class , the following lemma is required (see, e.g., [19, page 108]).

Lemma 9. *Let the function be given by
**
Then
** The functions and , or one of their rotations, show that both inequalities (36) and (37) are sharp. *

First we obtain sharp upper bounds for the Fekete-Szegö functional with real parameter.

Theorem 10. *Let be given by (1), and let be a real number. Then
**
where
**
and , are defined by (10) with and , respectively.**All estimates are sharp. *

*Proof. * Suppose that is given by (1). Then, from the definition of the class , there exist , such that
Set . Equating the coefficients of and on both sides of (41), we obtain
From (5), we have , and thus we obtain

It follows that

Making use of Lemma 9 (36), we have
or
where
Denote
where , , and .

Simple calculation shows that the function does not have a local maximum at any interior point of the open rectangle . Thus, the maximum must be attained at a boundary point. Since , , and , it follows that the maximal value of may be or .

Therefore, from (46), we obtain
where is given by (47).

Consider first the case . If , where is given by (39), then , and from (49), we obtain
which is the first part of the inequality (38). If , where is given by (40), then , and it follows from (49) that
and this is the third part of (38).

Next, suppose that . Then, , and thus, from (49), we obtain
which is the second part of the inequality (38).

In view of Lemma 9, the results are sharp for and or one of their rotations. From (41), we obtain that the extremal functions are and defined by (34) with and .

Next, we consider the Fekete-Szegö problem for the class with complex parameter.

Theorem 11. * Let be given by (1), and let be a complex number. Then,
**
The result is sharp. *

*Proof. * Assume that . Making use of (43), we obtain
The inequality (53) follows as an application of Lemma 9 (37) with
The functions and defined by (34) with and show that the inequality (53) is sharp.

Our Theorems 10 and 11 include several various results for special values of , , , and . For example, taking , in Theorem 10, we obtain the Fekete-Szegö inequalities for the class (see [2, 11]). The special case leads to the Fekete-Szegö inequalities for the class (see [2]). The Fekete-Szegö inequalities for the class (see [2]) are also included in Theorems 10 and 11.

#### Acknowledgments

The authors thank the referees for their valuable suggestions to improve the paper. The first and third authors’ research was supported by Atatürk University Rectorship under “The Scientific and Research Project of Atatürk University,” Project no. 2012/173.

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