tailieunhanh - báo cáo hóa học:" Research Article Normality of Composite Analytic Functions and Sharing an Analytic Function"

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 Normality of Composite Analytic Functions and Sharing an Analytic Function | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 417480 9 pages doi 2010 417480 Research Article Normality of Composite Analytic Functions and Sharing an Analytic Function Qifeng Wu 1 Bing Xiao 2 and Wenjun Yuan3 1 Shaozhou Normal College Shaoguan University Shaoguan 512009 China 2 Department of Mathematics Xinjiang Normal University Urumqi 830054 China 3 School of Mathematics and Information Science Guangzhou University Guangzhou 510006 China Correspondence should be addressed to Wenjun Yuan wjyuan1957@ Received 17 August 2010 Accepted 15 October 2010 Academic Editor Manuel De la Sen Copyright 2010 Qifeng Wu 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. A result of Hinchliffe 2003 is extended to transcendental entire function and an alternative proof is given in this paper. Our main result is as follows let a z be an analytic function F a family of analytic functions in a domain D and H z a transcendental entire function. If H f z and H g z share a z IM for each pair f z g z e F and one of the following conditions holds 1 H z - a z0 has at least two distinct zeros for any z0 e D 2 a z is nonconstant and there exists z0 e D such that H z - a z0 z - p0 pQ z has only one distinct zero p0 and suppose that the multiplicities l and k of zeros of f z - p0 and a z - a z0 at z0 respectively satisfy k Ip for each f z e F where Q p0 0 3 there exists a z0 e D such that H z - a z0 has no zero and a z is nonconstant then F is normal in D. 1. Introduction and Main Results Let f z and g z be two nonconstant meromorphic functions in the whole complex plane C and let a be a finite complex value or function. We say that f and g share a CM or IM provided that f - a and g - a have the same zeros counting or ignoring multiplicity. It is assumed that the reader .