Uniqueness Of Meromorphic Functions Concerning Their Derivatives And Related Problems

The uniqueness theory of meromorphic functions is an important part in the field of complex analysis. The so-called uniqueness is to study whether the functions satisfying certain conditions is unique or not. The uniqueness theory of meromorphic functions originates from some works in R. Nevanlinna. The Nevanlinna theory is important not only because it is the basis of modern meormorphic function theory, but also because it has quite an effect on the development of mathematical branches, and on the interaction among them. Especially, the Nevanlinna theory supplies a useful tool to the research of complex differential equations. In recent decades, many mathematicians paid more attention to it. Being an important part in the value distribution theory, the uniqueness theory of meromorphic functions also becomes very active in the world. Some foreign mathematicians such as F. Gross, W. Hayman, et al, have obtained hundreds of excellent results on it. Also, our mathematicians such as Xiong Qinglai, Yang Le and Yi Hongxun have proved lots of deep results on the research of the uniqueness theory. More and more new results appear as time goes on, which will make the theory rich and important.In this thesis, we consider the uniqueness theory of meromorphic functions concerning their derivates by using the value distribution theory as an important tool. The next, we will give five chapters on it.In chapter one, we briefly introduce some main concepts, fundamental results and usual notations in the value distribution theory of meromorphic functions. In chapter two, we study the uniqueness problem on entire functions concerning sharing a polynomial and extend some results by Fang.Theorem 2.1 Let f and g be two nonconstant entire functions, p ( z )be a polynomial of degree n1 , and let n1 , n ,k be three positive integers with share p ( z )CM, then either(1) , where c1 ,c2 and c are three constants satisfying 2 ( 1 2) 21(2) f ? tg, where t is a constant satisfying t n? 1. In chapter three, we study the uniqueness problem of entire functions concerning sharing a polynomial with their derivatives and obtain some results which improve some known theorems.Theorem 3.1 Let p ( z )be a transcendental entire function, Q( z )be a nonzero polynomial of degree q , and let be a positive integer. If f is a solution of the equation(and there exist some positive integer l such that , where E is a set of finite linear measure, then ? ( f) ??.Theorem 3.2 Let f ( z )be a non-constant entire function satisfying, where ? ( f) is not a positive integer, Q( z )a polynomial of degree and f ( m)share Q( z )CM, where n and m are positive integers satisfying , then there exist finite complex numbers satisfying such that where are certain finite complex constants. In chapter four, we consider the uniqueness problem of meromorphic functions concerning their derivatives sharing a small function and obtain some results which improve some known theorems.Theorem 4.1 Let f and ( 0,) a ?? ? be two meromorphic functions satisfying(,)(,),TraSrf? ) ) ( ,)(,()) p p E afEaLf ? .If If ifIn chapter five, we study the growth of solutions of a complex differential equation and prove the following results.Theorem 5.1 Let 0 h be a transcendental entire function with low growth and Q(z) be a nonzero polynomial, k ? 1be a positive integer and a be constants. If f(z)is any nonzero solution of the differential equation ( ) 1 0Theorem 5.2 Let f be an entire function of finite order. If f and differential polynomial share a CM, thenwhere c ? 0 is a constant.