tailieunhanh - Báo cáo hóa học: " Research Article On Harmonic Functions Defined by Derivative Operator"

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 Harmonic Functions Defined by Derivative Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 263413 10 pages doi 2008 263413 Research Article On Harmonic Functions Defined by Derivative Operator K. Al-Shaqsi and M. Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600 Selangor D. Ehsan Malaysia Correspondence should be addressed to M. Darus maslina@ Received 16 September 2007 Revised 20 November 2007 Accepted 26 November 2007 Recommended by Vijay Gupta Let SH denote the class of functions f h g that are harmonic univalent and sense-preserving in the unit disk U z z 1 where h z z y f 2akzk g z y f 1bkzk b1 1 . In this paper we introduce the class MH n X à of functions f h g which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class Mh n i a if fn z h gnE MH n x a where h z z- y f 2 akzk gn z -1 ny fcL1 bk zk and n E N0. Coefficient conditions such as distortion bounds convolution conditions convex combination extreme points and neighborhood for the class Mh n x a 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 complex domain C if both u and v are real harmonic in C. In any simply connected domain De C we can write f h g where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that h z g z l in D see 2 . Denote by SH the class of functions f h g that are harmonic univalent and sensepreserving in the unit disk U z z 1 for which f 0 h 0 fz 0 - 1 0. Then for f h g