Báo cáo hóa học: " Research Article Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators"

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 Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 147069 12 pages doi 2009 147069 Research Article Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators Nak Eun Cho Department of Applied Mathematics Pukyong National University Pusan 608-737 South Korea Correspondence should be addressed to Nak Eun Cho necho@ Received 3 March 2009 Accepted 1 May 2009 Recommended by Ramm Mohapatra The purpose of the present paper is to investigate some inclusion properties of certain classes of meromorphic functions associated with a family of linear operators which are defined by means of the Hadamard product or convolution . Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases. Copyright 2009 Nak Eun Cho. 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 A be the class of analytic functions in the open unit disk U z e C z 1 with the usual normalization f 0 f 0 - 1 0. If f and g are analytic in U we say that f is subordinate to g written f g or f z g z if there exists an analytic function w in U with w 0 0 and w z 1 for z e U such that f z g w z . Let N be the class of all functions Ộ which are analytic and univalent in U and for which U is convex with 0 1 and Re z 0 for z e U. We denote by S and K the subclasses of A consisting of all analytic functions which are starlike and convex respectively. Let M denote the class of functions of the form f z 1 akzk z k 0 which are analytic in the punctured open unit disk D U 0 . For 0 n p 1 we denote by MS n MK n and MC n p the subclasses of M consisting of all meromorphic functions which are respectively starlike

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