On Shared Values And Normal Families Of Meromorphic Functions

In this paper we study normal families of holomorphic functions and defects of more than two transcendental meromorphic functions concerning multiple values, which is an important subject in complex analysis. Much work was made on this respect. The present paper is divided into three parts.In the preface of Chapter 1, we give a review about the historical background of complex analysis and research achievements in these fields.In Chapter 1, we introduce some fundamental results, some definitions and some notations of Nevanlinna theory and defects theory.In Chapter 2, we obtained some results on normalities of holomorphic function families and its derivatives dealing shared values CM. and mainly prove Theorem 2.1.2 and Theorem2.4.2.Theorem 2.1.2 Let F be a family of holomorphic functions in a domain D , k (≥2) and p ( < k) be positive integers, and let K be a positive number. If, for each f∈F ,z∈D, f and f 'shared z^p CM ,and | f^(k)(z) |≤K whenever f (z) = z^p in D ,then F is normal in D .Theorem 2.4.2 Let F be a family of holomorphic functions in a domain D and k≥2 be a positive integer, K be a positive number and letα(z) be a polynomial of degree p. If, for each f∈F and z∈D, f and f 'sharedα(z)CM ,and | f^(k)(z) |≤K whenever f(z) -α(z)=0 in D . then F is normal in D .In Chapter three, we discuss relative quasi-defects of the common roots of more than two transcendental meromorphic functions concerning multiple values. We mainly prove Theorem3.1.1.Theorem 3.1.1 Let f₁, f₂,…, fl (l≥3) be l transcendental meromorphic functions, av ( v= 1,2,…,q) be finite, distinct, non-zero complex numbers, and k be a positive integer. Then...