Research Article | Open Access
Fekete-Szegö Problem for a New Class of Analytic Functions
We consider the Fekete-Szegö problem with complex parameter for the class () of analytic functions.
1. Introduction and Preliminaries
Let denote the class of functions of the form which are analytic in the open unit disk and denote the subclass of that are univalent in . A function in is said to be in class of starlike functions of order zero in , if for . Let denote the class of all functions that are convex. Further, is convex if and only if is star-like. A function is said to be close-to-convex with respect to a fixed star-like function if and only if for . Let denote of all such close-to-convex functions .
Fekete and Szegö proved a noticeable result that the estimate holds for any normalized univalent function of the form (1.1) in the open unit disk and for . This inequality is sharp for each (see ). The coefficient functional on normalized analytic functions in the unit disk represents various geometric quantities, for example, when , , becomes , where denote the Schwarzian derivative of locally univalent functions in . In literature, there exists a large number of results about inequalities for corresponding to various subclasses of . The problem of maximising the absolute value of the functional is called the Fekete-Szegö problem; see . In , Koepf solved the Fekete-Szegö problem for close-to-convex functions and the largest real number for which is maximised by the Koebe function is , and later in  (see also ), this result was generalized for functions that are close-to-convex of order .
Let be an analytic function with positive real part on with , which maps the unit disk onto a star-like region with respect to 1 which is symmetric with respect to the real axis. Let be the class of functions in for which and be the class of functions in for which where denotes the subordination between analytic functions. These classes were introduced and studied by Ma and Minda . They have obtained the Fekete-Szegö inequality for the functions in the class .
Motivated by the class in paper , we introduce the following class.
Definition 1.1. Let , . A function is in th class , if
where is defined the same as above.
If we set in (1.6), we get which is again a new class. We list few particular cases of this class discussed in the literature for was discussed recently by Swaminathan .The class for , where is considered in  (see also ).The class with was considered in  with reference to the univalencey of partial sums. whenever , the class considered in .
For geometric aspects of these classes, see the corresponding references. The class is new as the author Swaminathan  has introduced class which is subclass of the class , in his recent paper. To prove our main result, we need the following lemma.
2. Fekete-Szegö Problem
Our main result is the following theorem.
Theorem 2.1. Let , where with . If given by (1.1) belongs to , , then for any complex number The result is sharp.
Proof. If , then there exists a Schwarz function analytic in with and in such that Define the function by Since is a Schwarz function, we see that and . Define the function by In view of (2.2), (2.3), (2.4), we have Thus, From (2.4), we obtain Therefore, we have where Our result now is followed by an application of Lemma 1.2. Also, by the application of Lemma 1.2 equality in (2.1) is obtained when but Putting value of we get the desired results.
For class , Thus, putting and in Theorem 2.1, we get the following corollary.
Corollary 2.2. If given by (1.1) belongs to , then
- P. L. Duren, Univalent Functions, vol. 259 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1983.
- M. Fekete and G. Szegö, “Eine bemerkung uber ungerade schlichten funktionene,” Journal of London Mathematical Society, vol. 8, pp. 85–89, 1993.
- W. Koepf, “On the Fekete-Szegö problem for close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 101, no. 1, pp. 89–95, 1987.
- W. Koepf, “On the Fekete-Szegö problem for close-to-convex functions. II,” Archiv der Mathematik, vol. 49, no. 5, pp. 420–433, 1987.
- 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.
- W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Lang, and S. Zhang, Eds., pp. 157–169, International Press, Cambridge, Mass, USA.
- A. Swaminathan, “Sufficient conditions for hypergeometric functions to be in a certain class of analytic functions,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1578–1583, 2010.
- S. Ponnusamy and F. Ronning, “Integral transforms of a class of analytic functions,” Complex Variables and Elliptic Equations, vol. 53, no. 5, pp. 423–434, 2008.
- S. Ponnusamy, “Neighborhoods and Carathéodory functions,” Journal of Analysis, vol. 4, pp. 41–51, 1996.
- J. L. Li, “On some classes of analytic functions,” Mathematica Japonica, vol. 40, no. 3, pp. 523–529, 1994.
- A. Swaminathan, “Certain sufficiency conditions on Gaussian hypergeometric functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 4, article 83, 6 pages, 2004.
- 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.
- R. J. Libera and E. J. Złotkiewicz, “Coefficient bounds for the inverse of a function with derivative in ,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 251–257, 1983.
Copyright © 2011 Deepak Bansal. 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.