Meromorphic Functions With Differential Polynormials Sharing One Value

In 1920s, R. Nevanlinna introduced the characteristic functions of meromorphic functions and gave the famous Nevanlinna theory which is one of the greatest achieve-ments in mathematics in the 20th century. This theory is considered to be the basis of modern meromorphic function theory, and it has a very important effect on the develop-ment and syncretic of many mathematical branches such as diophantine approximation and non-Archimedean analysis. The theory is composed of two main theorems, which are called Nevanlinna's first and second theorems that had been significant breakthroughs in the development of the classic function theory, since the Nevanlinna's second the-orem generalizes and extends Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. From then on, Nevanlinna theory has been well developed in itself and widely applied to the researches of the uniqueness of meromorphic functions, normal families, complex dynamics and differential equations etc.In 1929, R. Nevanlinna applied the value distribution theory to consider the con-ditions under which a meromorphic function of a single variable could be determined and derived the famous Nevanlinna's five-value and four-value theorems. Since then, the research of meromorphic functions began. For over a half century, many foreign and domestic mathematicians have devoted themselves to the research and obtained lots of elegant results on the research of the uniqueness theory.The problem on meromorphic functions sharing values with their derivatives is the special and important case of uniqueness of meromorphic functions. In 1977, Rubel- Yang[36] considered the uniqueness on entire functions sharing two values CM with their derivatives. From then on, Mues-Steinmetz[35], L.Z. Yang[44], Gundersen[14], Frank-Weissenborn[11] etc. improved and proved relative results. After Raider Briick imposed Briick conjecture in 1996, many foreign and domestic mathematicians began to research the problem on meromorphic functions sharing one value with their derivatives and proved many results. Recently, Fang-Hua[7], Zhang-Lin[51], Q.C.Zhang[49] gave some results about meromorphic functions and their differential polynomials sharing one value.In this paper, we will give some results on the uniqueness of meromorphic functions with their differential polynomials sharing one value under the guidance of Professor Hu peichu. This thesis consists of three chapters.In chapter 1, we briefly introduce the background of this thesis, which contains some fundamental results and notations of Nevanlinna theory.In chapter 2, we study the uniqueness problems on entire functions sharing one value. We improve and generalize some previous results of Zhang xiaoyu and Lin weichuan[51]. The main result is the following:Theorem 2.1. Let f(z) and g(z) be two nonconstant entire functions. Let P(f)= and n,k, m be three positive integers with share 1 CM, then (1) If 0≤i< m, then either f(z)≡g(z) or f, g satisfy the algebraic equation R(f, g)≡0, where (2) If i=m. then either f(z)≡tg(z), where t is a constant satisfying where c1, c2 and c are three constants satisfyingWith regard to sharing value IM, If n,κ, m satisfy n+m> (5k+7)(m+1), we also have the following result:Theorem 2.2. Let f(z) and g(z) be two nonconstant transcendental entire functions. and n, k, m be three positive integers with share 1 IM, then (1) If 0≤i< m, then f and g satisfy the algebraic equation R(f,g)1≡0, where (2) If i= m, then f(z)= tg(z), where t is a constant satisfying g(z)= c2e-cz, where c1,c2 and c are three constants satisfyingIn chapter 3, we study the uniqueness problem on meromorphic functions concern-ing differential polynomials that share one value ignoring multiplicity. Moreover, we greatly generalize the main result obtained by Zhang, Chen and Lin[50]. The main re-sult is the following:Theorem 3.1. Let f and g be two transcendent meromorphic functions, let n and m be two positive integers with n> 11m+22, and letα1z+α0. whereα0(≠0),α1,…,αm-1,αm(≠0) are complex constants. If fnP(f)f' and gnP(g)g'share 1 IM, then either f≡tg for a constant t such that td= 1, where d = (n+m+1,…, n+m+1 - i,…, n+1), am_i≠0 for some i= 0,1,…, m, or f and g satisfy the algebraic equation R(f,g)≡0, whereIn addition, we can also change the nature of the sharing value 1 with the concept of weighted sharing which was introduced by I. Lahiri.Theorem 3.2. Let f and g be two nonconstant meromorphic functions, let n and m be two positive integers with n> max{m+10,3m+3}, and let…+α1z+α0, whereα0(≠0),α1,…,αm-1, αm(≠0) are complex constants. If fnP(f)f' and gnP(g)g'share (1,2), then the conclusion of Theorem 3.1 holds.Naturally, we ask the following question: In Theorem 3.1 can the nature of sharing the value 1 be further relaxed other than the concept of weighted sharing?We now state Theorem 3.3 which answers the above question.Theorem 3.3. Let f and g be two nonconstant meromorphic functions, let n and m be two positive integers with n> max{m+10,3m+3}, and let P(z) =αmzm + whereα0(≠0),α1,…,αm-1,αm(≠0) are complex constants. If fnP(f)f' and gnP(g)g'satisfy E3)(1, fnP(f)f)= E3)(1gnP(g)g'), then the conclusion of Theorem 3.1 holds.