Two Unicity Theorems Of Entire Functions

In 1920s, R. Nevanlinna introduced the characteristic functions of meromorphic functions and gave the famous Nevanlinna value distribution theory which is one of the greatest achievements in mathematics in the 20^th century. This theory is considered to be the basis of modern meromorphic function theory, and it has a very importanteffect on the development and syncretic of many mathematical branches. Especially, Nevanlinna theory has been used as a very powerful tool and has made the research field more active and popular after it was successfully applied to the research of the global analytic solutions of complex differential equations. Taking advantage of the theory just made by himself, R. Nevanlinna studied the conditions with which a meromorphic function can be determined and obtained two celebrated unicity theorems on meromorphic functions, which are called Nevanlinna's five-value theorem and four-value theorem. Since then, the research of the unicity theorem began.For over a half century, many foreign and domestic mathematicians, such as F. Gross, M. Ozawa, G Frank, E. Mues, N. Steinmetz, H. Ueda, G Gundersen, Xiong Qinglai and Yang Le, have devoted themselves to the research and obtained lots of elegant results on the research of the value distribution theory.The paper's result is related with the unicity theorem. Because of its value and the relationship with the normal families, the unicity theorem needs more creative work. This thesis consists of four chapters:In chapter 1, we will briefly introduce the history of the unicity theorem.In chapter 2, we will briefly introduce some fundamental results, definitions and some notations.In chapter 3, we mainly study the entire functions that share a rational function with their derivatives. In 2006, Chang and Fang proved the following theorem:Theorem 1 Let f be a nonconstant entire function,(?) a nonzero finite complex number, k and m two distinct positive integers. If f,f^(k),f^(m) share (?) CM,then f^(k)=f^(m).We prove the following result:Theorem 2 Let f be a transcendental entire function, R a nonzero rational function, k and m two distinct positive integers. If f,f^(k),f^(m) share R CM,then f^(k)=f^(m).In chapter 4, we mainly study the entire functions that share a small function with their derivatives. G Gundersen and Yang L. Z. proved the following theorem:Theorem 3 Let f be an entire function of finite order and (?) a complex number. If f and f' share (?) CM, then f - (?) = c(f'-(?)) for some non-zero constant c.We prove that the theorem 3 remains valid if the value (?) is replaced by a small function:Theorem 4 Let f be an entire function of finite order and (?) an entire function of order less than f's. If f and f' share (?) CM,then f-(?) = c(f'-a) for some non-zero constant c.In this paper, an "entire function" will mean a function that is analytic in the whole complex plane, a "meromorphic function" will mean a single-valued analytic function that has not any other singularity except poles in the whole complex plane.We say that two functions (two entire functions or two meromorphic functions) f and g share a function (?) CM (counting multiplicities), if f-(?) and g-(?) have the same zeros with the same multiplicities. For the case that (?) is constant,(?) is also called a CM-shared value of f and g.