About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 415319, 7 pages
http://dx.doi.org/10.1155/2013/415319
Research Article

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 Brasov, Iuliu Maniu 50, 50091 Brasov, 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 Kiliçman

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 Špacek [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 [612].

For a function , we consider the following differential operator introduced by Raducanu 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 Salagean [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.

References

  1. R. J. Libera, “Univalent α-spiral functions,” Canadian Journal of Mathematics., vol. 19, pp. 725–733, 1967. View at MathSciNet
  2. F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, pp. 8–12, 1969. View at MathSciNet
  3. L. Špaček, “Contribution à la théorie des functions univalents,” Casopis Pro Pestování Matematiky A Fysiky, vol. 62, no. 2, pp. 12–19, 1932.
  4. M. Fekete and G. Szegö, “Eine bemerkung uber ungerade schlichte funktionen,” The Journal of the London Mathematical Society, vol. 8, no. 2, pp. 85–89. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Pfluger, “The Fekete-Szegő inequality for complex parameters,” Complex Variables. Theory and Application, vol. 7, no. 1–3, pp. 149–160, 1986. View at MathSciNet
  6. E. Deniz and H. Orhan, “The Fekete-Szegö problem for a generalized subclass of analytic functions,” Kyungpook Mathematical Journal, vol. 50, no. 1, pp. 37–47, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  7. E. Deniz, M. Çağlar, and H. Orhan, “The Fekete-Szegö problem for a class of analytic functions defined by Dziok-Srivastava operator,” Kodai Mathematical Journal, vol. 35, no. 3, pp. 439–462, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. K. Mishra and P. Gochhayat, “Fekete-Szegö problem for a class defined by an integral operator,” Kodai Mathematical Journal, vol. 33, no. 2, pp. 310–328, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  9. H. Orhan, E. Deniz, and D. Raducanu, “The Fekete-Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 283–295, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  10. H. Orhan, E. Deniz, and M. Çağlar, “Fekete-Szegö problem for certain subclasses of analytic functions,” Demonstratio Mathematica, vol. 45, no. 4, pp. 835–846, 2012.
  11. H. M. Srivastava, A. K. Mishra, and M. K. Das, “The Fekete-Szegő problem for a subclass of close-to-convex functions,” Complex Variables. Theory and Application, vol. 44, no. 2, pp. 145–163, 2001. View at MathSciNet
  12. P. Wiatrowski, “The coefficients of a certain family of holomorphic functions,” Zeszyty Naukowe Uniwersytetu Lodzkiego Nauki Matematyczno Przyrodniczego Seria, no. 39, pp. 75–85, 1971. View at MathSciNet
  13. D. Răducanu and H. Orhan, “Subclasses of analytic functions defined by a generalized differential operator,” International Journal of Mathematical Analysis, vol. 4, no. 1–4, pp. 1–15, 2010. View at MathSciNet
  14. G. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—5th Romanian-Finnish seminar, vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  15. F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,” International Journal of Mathematics and Mathematical Sciences, no. 25–28, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. Murugusundaramoorthy, “Subordination results for spiral-like functions associated with the Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 23, no. 2, pp. 97–103, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. O. S. Kwon and S. Owa, “The subordination theorem for λ-spirallike functions of order α,” Sūrikaisekikenkyūsho Kōkyūroku, no. 1276, pp. 19–24, 2002. View at MathSciNet
  18. H. Silverman, “Sufficient conditions for spiral-likeness,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 4, pp. 641–644, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Z. Nehari, Conformal Mapping, McGraw-Hill, London, UK, 1952. View at MathSciNet