| In the 1920's,the mathematician Rolf Nevanlinna,a Finlander. founded one of the most important theories of the twentieth century,the value distribution theory of meromorphic functions over the open complex plane C.Which is usually called Nevanlinna theory in honor of him.This tmeory is composed of two main theo-rys.Which are called Nevanlinna's first and second main theorys that had been sig-nificant breakthroughs in the development of the classical function theory. Since the latter generalizes and extends Picard's first theorem greatly,and hence it denoted the beginning of the theory of meromorphic functions.For some eighty years,Nevanlinna theory has been widely applied to the researches of the unicity of meromorphic functions,normal families,complex dynamics and differential equations etc.In 1929,Rolf Nevanlinna applied his value distibution theory to consider the conditions under which a meromorphic function of a single variable could be de-termined,and derived the famous Nevanlinna's five-value,four-value and three-value theories. 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 theoty.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[4]considered the uniqueness on entire functions sharing two values CM with their derivatives. From then on,Mues-Steinmetz[25],L.Z.Yang[26],Gundersen[27]etc. improved relative results.After Raider Bruck imposed Bruck 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, Q.C.zhang[24],Fang-Hua[23],Zhang-Lin[21]gave some results about meromorphic func-tions and their differential polynomials sharing one value.The uniqueness of meromorphic function mainly studies conditions under which there is only one function.We know that the contions used determine a transcenden-tal meromorphic function is different from that of a polynomial.Therefore,complicated and interesting.In recent decades,many mathematicians paid close attention to it,which makes it become a very avtive subject all around the word.In chapter l.we briefly introduce the background of this thesis,which contains some fundamental resuits and notations of Nevanlinna theory.In chapter 2,we study the uniqueness problems on meromorphic functions shar-ing one value.We improve and generalize some previous results of R.S.Dyavanal[9].The main result is the following.Theorem 2.1. Let f(z) and g(z) be two non-constant meromorphic functions,whose zeros and poles are of multiplities at least s,where s is a positive integer.Let n≥2 be an integer satisfying (n+1)s≥21. If fn f' and gn g' share the 1 IM,then either f(?)dg,for some (n+1)-th root of unity d or f(z)=c1ecx and g(z)= c2e-cz,where c,c1,c2 are contants satisfying(c1c2)n+1c2=-1.When we consider the differential polynomial of fn(f-1)f' and gn(g-1)g',then we get the following theory.Theorem 2.2. Let f(z) and g(z) be two non-constant meromorphic functions,whose zeros and poles are of multiplities at least s,where s is a positive integer andθ(∞,f)>2/(n+1).Let n≥2 be an integer satisfying (n+1)s≥21. If fn(f-1)f' and gn(g-1)g' share the 1 IM,then f(z)(?)g(z). Smillarly we concider the differential polynomial of fn(f-1)2 f' and gn(g-1)2g' ,then we get the following: Theorem 2.3.Let f(z)and g(z)be two non-constant meromorphic functions,whose zeros and poles are of multiplities at least s,where s is a positive integer.Let n≥2 be an integer satisfying(n+1)s≥21.If fn(f-1)2 f' and gn(g-1)2g' share the 1 IM,then f(z)≡g(z).In chapter 3,we study the uniqueness problem on meromorphic funcotions con-cerning differential polynomials that share one value ignoring multiplicity.More-over,we greatly generalize the main result obtained by Lu,chen and Yi[12]. The main result is following:Theorem 3.1.Let f(z) and g(z)be non-constant meromorphic functions,n,k,be two integers satisfying n≥k+3.Then [fnP(f)](k)=1(am≠0)has infinitely many solutions.Theorem 3.2.Let f(z) and g(z)be non-constant meromorphic functions.Let P(f)= amfm+am-1fm-1+…+aifi(am≠0,ai≠0,0≤i≤m),and n,k,m,be three positive integers with n>6m+9k+13.If[fnP(f)](k) and [gnP(g)](k) share 1 IM,f and g share∞IM,then(1)If 0≤i<m,then either f(z)≡g(z) or f,g satisfy the algebraic equation R(f,g)≡0,where R(w1,w2)=wn1P(w1)-w2P(w2).(2)If i=m,then either f(z)≡g(z),where t is a constant sat,isfying ln+m=1 or f(z)=c1ecz and g(z)=c2e-cz,where c,c1,c2 are contants satisfying... |
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