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Báo cáo hóa học: "Research Article Univalence of Certain Linear Operators Defined by Hypergeometric Function"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Univalence of Certain Linear Operators Defined by Hypergeometric Function | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 807943 12 pages doi 10.1155 2009 807943 Research Article Univalence of Certain Linear Operators Defined by Hypergeometric Function R. Aghalary and A. Ebadian Department of Mathematics Faculty of Science Urmia University Urmia Iran Correspondence should be addressed to A. Ebadian a.ebadian@urmia.ac.ir Received 11 January 2009 Accepted 22 April 2009 Recommended by Vijay Gupta The main object of the present paper is to investigate univalence and starlikeness of certain integral operators which are defined here by means of hypergeometric functions. Relevant connections of the results presented here with those obtained in earlier works are also pointed out. Copyright 2009 R. Aghalary and A. Ebadian. 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. 1. Introduction and Preliminaries Let H denote the class of all analytic functions f in the unit disk D z e C z 1 . For n 0 a positive integer let An p eH f z z TO an kZn k k 1 1.1 with A1 A where A is referred to as the normalized analytic functions in the unit disc. A function f eA is called starlike in D if f D is starlike with respect to the origin. The class of all starlike functions is denoted by S S 0 . For a 1 we define S a i f eA Re zf S f eA R fz 2 a z e D 1.2 and it is called the class of all starlike functions of order a. Clearly S a c S for 0 a 1. For functions fj z given by TO fj z s ak jzk j 12 k 0 1.3 2 Journal of Inequalities and Applications we define the Hadamard product or convolution of fi z and f2 z by TO f1 f2 z ak 1 ak 2zk f2 f1 z . 1.4 k 0 An interesting subclass of S the class of all analytic univalent functions is denoted by U a F A and is defined by U a p A p eA _ z V _ z V 1 1 1 -a f af f z - 1 A z e D 1.5 where 0 a 1 0 n an and A 0. The special