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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 415319, 7 pages
Coefficient Estimates and Other Properties for a Class of Spirallike Functions Associated with a Differential Operator
1Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey
2Department of Mathematics, Faculty of Mathematics and Computer Science, Transylvania University of Braşov, Iuliu Maniu 50, 50091 Braşov, Romania
3Department 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.
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.
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 .
Note that is the class of spirallike functions introduced by Špaček , 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 ) 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 ) 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 :(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 ;(ii), the operator studied by Al-Oboudi .
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 .
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.
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 ). Let . If where , , then .
Corollary 7 (see ). 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.
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]).
First we obtain sharp upper bounds for the Fekete-Szegö functional with real parameter.
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 ). The Fekete-Szegö inequalities for the class (see ) are also included in Theorems 10 and 11.
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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