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Báo cáo hóa học: " Research Article Normality Criteria of Lahiri’s Type and Their Applications"

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 Criteria of Lahiri’s Type and Their Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 873184 16 pages doi 10.1155 2011 873184 Research Article Normality Criteria of Lahiri s Type and Their Applications Xiao-Bin Zhang 1 Jun-Feng Xu 1 2 and Hong-Xun Yi1 1 Department of Mathematics Shandong University Jinan Shandong 250100 China 2 Department of Mathematics Wuyi University Jiangmen Guangdong 529020 China Correspondence should be addressed to Jun-Feng Xu xujunf@gmail.com Received 22 September 2010 Revised 9 January 2011 Accepted 9 February 2011 Academic Editor Siegfried Carl Copyright 2011 Xiao-Bin Zhang 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. We prove two normality criteria for families of some functions concerning Lahiri s type the results generalize those given by Charak and Rieppo Xu and Cao. As applications we study a problem related to R. Bruck s Conjecture and obtain a result that generalizes those given by Yang and Zhang Lu Xu and Chen. 1. Introduction and Main Results Let c denote the complex plane and let f z be a nonconstant meromorphic function in c. It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic function T r f the proximity function mfr f the counting function Nfr f see e.g. 1-4 and S r f denotes any quantity that satisfies the condition Sfr f o T r f as r TO outside of a possible exceptional set of finite linear measure. A meromorphic function ajz is called a small function with respect to f z provided that T fr a S r f . Let f z and g z be two nonconstant meromorphic functions. Let ajz and bjz be small functions of f z and g z . f z afz g z b z means f z - afz and g z -bjz have the same zeros counting multiplicity and f z TO g z TO means that f and g have the same poles .