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Báo cáo hóa học: " Research Article Subordination and Superordination on Schwarzian Derivatives"

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 Subordination and Superordination on Schwarzian Derivatives | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 712328 18 pages doi 10.1155 2008 712328 Research Article Subordination and Superordination on Schwarzian Derivatives Rosihan M. Ali 1 V. Ravichandran 2 and N. Seenivasagan3 1 School of Mathematical Sciences Universiti Sains Malaysia USM 11800 Penang Malaysia 2 Department of Mathematics University of Delhi Delhi 110 007 India 3 Department of Mathematics Rajah Serfoji Government College Thanjavur 613 005 India Correspondence should be addressed to Rosihan M. Ali rosihan@cs.usm.my Received 4 September 2008 Accepted 30 October 2008 Recommended by Paolo Ricci Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1 z zf z f z q2 z or q1 z 1 zf z f z q2 z . As applications sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of f . Copyright 2008 Rosihan M. Ali 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. 1. Introduction Let H U be the class of functions analytic in U z e C z 1 and H a n be the subclass of H U consisting of functions of the form f z a anzn an 1zn 1 . We will write H H 1 1 . Denote by A the subclass of H 0 1 consisting of normalized functions f of the form f z z jrakzk z e U . k 2 1.1 Let S and K respectively be the familiar subclasses of A consisting of starlike and convex functions in U. The Schwarzian derivative f z of an analytic locally univalent function f is defined by f z rz f z 1 f z 2 A f 1.2 2 Journal of Inequalities and Applications Owa and Obradovic 1 proved that if f e A satisfies r 1 zf z 2 1 RIK1 - z f z 0 1.3