International Scholarly Research Notices

International Scholarly Research Notices / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 861671 | 4 pages | https://doi.org/10.1155/2014/861671

On the Fekete-Szegö Problem for a Class of Analytic Functions

Academic Editor: Y. Han
Received02 Dec 2013
Accepted21 Jan 2014
Published04 Mar 2014

Abstract

Let denote the class of functions which are analytic in the unit disk and given by the power series . Let be the class of convex functions. In this paper, we give the upper bounds of for all real number and for any in the family , Re for  some .

1. Introduction

Let denote the class of functions which are analytic in the unit disk and satisfy . The set of all functions that are univalent will be denoted by . Let and be the classes of convex, starlike of order and close-to-convex functions, respectively. Fekete and Szegö [1] proved that holds for any and that this inequality is sharp. The coefficient functional on in plays an important role in function theory. For example, , where is the Schwarzian derivative. The problem of maximizing the absolute value of the functional is called the Fekete-Szegö problem. In the literature, there exist a large number of results about the Fekete-Szegö problem (see, for instance, [211]).

For and , let denote the class of functions satisfying and for some . Al-Abbadi and Darus [7] investigated the Fekete-Szegö problem on the class .

Let be the class of functions in satisfying the inequality for some function . In [11], Srivastara et al. studied the Fekete-Szegö problem on the class for by proving that Srivastara et al. held that the inequality (5) was sharp. However, the extremal function given in [11] did not exist in the case of .

In this paper, we solve the Fekete-Szegö problem for the family As a corollary of the main result, we find the sharp upper bounds for absolute value of the Fekete-Szegö functional for the class defined by Clearly, is a subclass of . In the case of , we get sharp estimation of the absolute value of the Fekete-Szegö functional for the class and for all real number , which prove that the inequality (5) is not sharp actually when .

2. Main Result

Let be the class of functions that are analytic in and satisfy for all . The following two lemmas can be found in [2].

Lemma 1 (see [2]). If is in the class , then, for any complex number , one has . The inequality is sharp.

Lemma 2 (see [2]). If is in the class and is a complex number, then . The inequality is sharp.

Theorem 3. If is in the class and is a real number, then

Proof. By definition, is in the class if and only if there exists a function such that . A simple computation shows . Thus, So, by Lemmas 1 and 2, we have Putting and , we get from (10) that , where Since and , we will calculate the maximum value of for .
Case 1. Suppose . Then it follows from (11) that Since does not have a local maximum at any point of the open rectangle . Hence, must attain its maximum at a boundary point. Since for and we have
Case 2. Suppose . Then, we get from (11) that Since must attain its maximum at a boundary point of the rectangle .
Since we get
Case 3. Suppose . Then, (11) gives
Since must attain its maximum at a boundary point of the rectangle . Since it follows that
Combining (15), (19) with (23), we get (8). Since inequalities in Lemmas 1 and 2 are sharp, it follows that inequality (8) is also sharp. The proof is completed.

Since if and only if , by a simple calculation, we have the following.

Corollary 4. If , then

Letting in (24), we get the following.

Corollary 5. If , then

Remark 6. In [11], Srivastava et al. gave a function satisfying , where and Srivastava held that and when . But implies that . So satisfying the above conditions does not exist.

Conflict of Interests

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

Acknowledgment

The paper is supported by Educational Commission of Hubei Province of China (D2011006).

References

  1. M. Fekete and G. Szegö, “Eine bemerkung über ungerade schlichte funktionen,” The Journal of the London Mathematical Society, vol. 8, no. 2, pp. 85–89, 1933. View at: Publisher Site | Google Scholar | 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: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. H. R. Abdel-Gawad and D. K. Thomas, “The Fekete-Szegö problem for strongly close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 114, no. 2, pp. 345–349, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. R. R. London, “Fekete-Szegö inequalities for close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 117, no. 4, pp. 947–950, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. N. E. Cho, “On the Fekete-Szegö problem and argument inequality for strongly quasi-convex functions,” Bulletin of the Korean Mathematical Society, vol. 38, no. 2, pp. 357–367, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. S. Kanas and H. E. Darwish, “Fekete-Szegö problem for starlike and convex functions of complex order,” Applied Mathematics Letters, vol. 23, no. 7, pp. 777–782, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. M. H. Al-Abbadi and M. Darus, “The Fekete-Szegö theorem for a certain class of analytic functions,” Sains Malaysiana, vol. 40, no. 4, pp. 385–389, 2011. View at: Google Scholar
  8. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. K. Al-Shaqsi and M. Darus, “On Fekete-Szegö problems for certain subclass of analytic functions,” Applied Mathematical Sciences, vol. 2, no. 9-12, pp. 431–441, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. B. Bhowmik, S. Ponnusamy, and K.-J. Wirths, “On the Fekete-Szegö problem for concave univalent functions,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 432–438, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. H. M. Srivastava, A. K. Mishra, and M. K. Das, “The Fekete-Szegö problem for a subclass of close-to-convex functions,” Complex Variables and Elliptic Equations, vol. 44, no. 2, pp. 145–163, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2014 Zhigang Peng. 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.


More related articles

801 Views | 404 Downloads | 1 Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.