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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 for Higher-Order Derivatives of Multivalent Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 830i38 12 pages doi 10.1155 2008 830138 Research Article Subordination for Higher-Order Derivatives of Multivalent Functions Rosihan M. Ali 1 Abeer O. Badghaish 1 2 and V. Ravichandran3 1 School of Mathematical Sciences Universiti Sains Malaysia USM 11800 Penang Malaysia 2 Mathematics Department King Abdul Aziz University P.O. Box 581 Jeddah 21421 Saudi Arabia 3 Department of Mathematics University of Delhi Delhi 110 007 India Correspondence should be addressed to Rosihan M. Ali rosihan@cs.usm.my Received 18 July 2008 Accepted 24 November 2008 Recommended by Vijay Gupta Differential subordination methods are used to obtain several interesting subordination results and best dominants for higher-order derivatives of p-valent functions. These results are next applied to yield various known results as special cases. 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. Motivation and preliminaries For a fixed p e N 1 2 . let Ap denote the class of all analytic functions of the form TO f z z 2 a.k pzk v k 1 1.1 which are p-valent in the open unit disc U z e C z 1 and let A A1. Upon differentiating both sides of 1.1 q-times with respect to z the following differential operator is obtained f Az Vp q zp-q jr X k p q ak pzk p-q 1.2 where à p q p p q p e N q e Nu 0 . p - q 1.3 2 Journal of Inequalities and Applications Several researchers have investigated higher-order derivatives of multivalent functions see for example 1-10 . Recently by the use of the well-known Jack s lemma 11 12 Irmak and Cho 5 obtained interesting results for certain classes of functions defined by higher-order derivatives. Let f and g be analytic in U. Then f is subordinate to g written as f z g z z e U if there is
