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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 127615, 7 pages
Fekete-Szegő Inequality for a Subclass of -Valent Analytic Functions
1Department of Mathematics, Government College University Faisalabad, Faisalabad, Punjab 38000, Pakistan
2Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa 23200, Pakistan
3School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D. Ehsan, Malaysia
Received 18 January 2013; Accepted 11 April 2013
Academic Editor: Mina Abd-El-Malek
Copyright © 2013 Mohsan Raza 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.
The main object of this paper is to study Fekete-Szegő problem for the class of -valent functions. Fekete-Szegő inequality of several classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.
1. Introduction and Preliminaries
Let denote the class of functions of the form which are analytic in the open unit disk . Also, , the usual class of analytic functions defined in the open unit disk . Let and be analytic in . We say that the function is subordinate to the function and write , if and only if there exists Schwarz function , analytic in such that , for , and . In particular, if is univalent in , then we have the following equivalence:
For any two analytic functions of the form (1) and with the convolution (Hadamard product) is given by Let be an analytic function with positive real part on with , , which maps the unit disk onto a region starlike with respect to which is symmetric with respect to the real axis. Denote by the class of functions analytic in for which The class was defined and studied by Ali et al. . They obtained the Fekete-Szegő inequality for functions in the class . The class coincides with the class discussed by Ma and Minda . Owa  introduced a subclass of -valently Bazilevic functions . A function is said to be in the class if and only if where , , and . We now define the following subclass of analytic functions.
Definition 1. Let be a univalent starlike function with respect to 1 which maps the unit disk onto a region in the right half-plane which is symmetric with respect to the real axis with and . A function is in the class if where , , and .
Definition 2. A function is in the class if
where , , and In other words, a function is in the class if .
We have the following special cases.(i) coincides with the class introduced and studied by Ali et al. .(ii)For , , and , we have the class introduced and studied by Ravichandran et al. .(iii)For and , the class reduces to introduced and studied by Owa .(iv)For , the class reduces to the class defined as (v)For , the class is defined as (vi) is investigated by Ma and Minda .(vii)For , the class reduces to the class studied by Chen .
We need the following results to obtain our main results.
Lemma 3 (see ). Let be the class of analytic functions normalized by satisfying condition If and , then For or the equality holds, if and only if or one of its rotation. For the equality holds, if or one of its rotation. the equality holds for , if and only if or one of its rotation, while for , the equality holds, if and only if or one of its rotation. The above upper bound for is sharp, and it can be improved as follows:
Lemma 4 (see [6, (7), page 10]). If and , then for any complex number . The result is sharp for the functions or .
Lemma 5 (see ). If , then for any real number and , the following sharp estimate holds: where The extremal function up to the rotations is of the form The sets , are defined as follows:
2. Main Results
Theorem 6. Let , where , are real with and . Let If is of the form (1) and belongs to the class , then Furthermore, for , and for , For any complex number , Also, where is defined in Lemma 3 and These results are sharp.
Proof. Since , therefore we have for a Schwarz function such that Now, Also, we have Comparing the coefficients of , , and after simple calculations, we obtain where and are defined in (25). It can be easily followed from (30) that where The results from (20) to (22)are obtained by using Lemma 3, (23)by using Lemma 4, and (24) by using Lemma 5. To show that these results are sharp, we define the functions , , and such that with It is clear that the functions . Let . If or , then the equality occurs for the function or one of its rotations. For , the equality is attained, if and only if is or one of its rotations. When , then the equality holds for the function or one of its rotations. If , then the equality is obtained for the function or one of its rotations.
2.1. Application of Theorem 6 to the Function Defined by Convolutions
Theorem 11. Let , where , are real with and . Let If is of the form (1) and belongs to the class , then Furthermore, for , and for , For any complex number , Also, where is defined in Lemma 5 and These results are sharp.
Proof. Since , therefore we have for a Schwarz function such that Now, Also, we obtain Comparing the coefficients of , , and after simple calculations, we obtain The remaining proof of the theorem is similar to the proof of Theorem 6.
The work here is fully supported by LRGS/TD/2011/UKM/ICT/03/02.
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