tailieunhanh - Báo cáo hóa học: " Research Article Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product"

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 Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 759251 12 pages doi 2009 759251 Research Article Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product Om P. Ahuja 1 H. Ozlem Giiney 2 and F. Muge Sakar2 1 Department of Mathematical Sciences Kent State University 14111 Claridon-Troy Road Burton OH 44021 USA 2 Department of Mathematics Faculty of Science and Arts Dicle University 21280 Diyarbakir Turkey Correspondence should be addressed to H. Ozlem Guney ozlemg@ Received 25 February 2009 Accepted 12 September 2009 Recommended by Yong Zhou We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product. Copyright 2009 Om P. Ahuja 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 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 D c C we can write f h g where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. Note that f h g reduces to h if the coanalytic part g is zero. For p 1 denote by H p the set of all multivalent harmonic functions f h g defined in the open unit disc U where h and g defined by TO TO h z zp y anZn g z y bnZn bp t-1 1 t e N 1 2 . n p t n p t-1 are analytic functions in U. Let F be a fixed multivalent harmonic function given by TO TO F z H z G Z zp y An zn y Bn zn Bp t-1 1 t e N. n p t n p t-1 2 Journal of Inequalities and Applications A function f e H p is said to be in the class HF p t a k if R zf F z z f F z J zf F z z f F z p k pa where f F is a harmonic convolution of f and F. Note that z d d0 r eie f z d dd f r eie . Using the fact Re w k w - p pa