tailieunhanh - Báo cáo hóa học: " Research Article Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method"

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 Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 896087 6 pages doi 2010 896087 Research Article Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method Yaẹar Polatoglu H. Esra Ozkan and Emel Yavuz Duman Department of Mathematics and Computer Science Istanbul Kultur University Istanbul 31456 Turkey Correspondence should be addressed to H. Esra Ozkan Received 3 June 2010 Accepted 26 July 2010 Academic Editor N. Govil Copyright 2010 Yasar Polatoglu 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. Any harmonic function in the open unit disc D z z 1 can be written as a sum of an analytic and antianalytic functions f h z g z where h z and g z are analytic functions in D and are called the analytic part and the coanalytic part of f respectively. Many important questions in the study of the classes of functions are related to bounds on the modulus of functions growth or the modulus of the derivative distortion . In this paper we consider both of these questions. 1. Introduction Let U be a simply connected domain in the complex plane. A harmonic function f has the representation f h z g z where h z and g z are analytic in U and are called the analytic and coanalytic parts of f respectively Let h z a0 a1z a2z2 and g z b0 b1z b2 z2 be analytic functions in the open unit disc D. If Jf z h z 2- g z 2 0 then f h z g z is called the sense-preserving harmonic univalent function in D. The class of all sense-preserving harmonic univalent functions is denoted by SH with a0 b0 0 a1 1 and b1 1 and the class of all sense-preserving harmonic univalent functions is denoted by S H with a0 b0 b1 0 a1 1. For convenience we will examine sensepreserving functions that is functions for .