tailieunhanh - Báo cáo hóa học: " Research Article On Meromorphic Harmonic Functions with Respect to k-Symmetric Points"

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 On Meromorphic Harmonic Functions with Respect to k-Symmetric Points | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 259205 11 pages doi 2008 259205 Research Article On Meromorphic Harmonic Functions with Respect to k-Symmetric Points K. Al-Shaqsi and M. Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi Selangor D. Ehsan 43600 Malaysia Correspondence should be addressed to M. Darus maslina@ Received 22 May 2008 Revised 20 July 2008 Accepted 23 August 2008 Recommended by Ramm Mohapatra In our previous work in this journal in 2008 we introduced the generalized derivative operator Di for f e SH. In this paper we introduce a class of meromorphic harmonic function with respect to k-symmetric points defined by Djm. Coefficient bounds distortion theorems extreme points convolution conditions and convex combinations for the functions belonging to this class are obtained. Copyright 2008 K. Al-Shaqsi and M. Darus. 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 A continuous function f u iv is a complex valued harmonic function in a domain D c C if both u and v are real harmonic in D. In any simply connected domain we write f h g where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that h g in D see 1 . Hengartner and Schober 2 investigated functions harmonic in the exterior of the unit disk U z z 1 . They showed that complex valued harmonic sense preserving univalent mapping f must admit the representation f z h z g z A logIzl C1-1 where h z and g z are defined by h z az y UnZ n g z fiz y b-zf for 0 a A e C and z e U. 2 Journal of Inequalities and Applications For z e U 0 let MH denote the class of functions TO. ỌỌ anZn y bnZn f z htz g z z y n 1 which are .