Uniqueness Of A Class Of Special Entire Function And Meromorphic Function Solutions Of Fermat-type Function Equations

In this thesis,we mainly study two aspects of meromorphic function theory,based on the value distribution theory established by R.Nevanlinna.On the one hand,we study the uniqueness theorem of non-constant entire functions.Firstly,we study a class of special entire function of small order,discuss the nonobvious sufficient conditions for the non-constant entire functio n f（z） to be an even function,the following conclusion is obtained.（1）Let f（z）be an non-constant entire function,with order ρ_f<1,if f（z） and f（-z）have two finite IM sharing values,than f（z）is even function,i.e.f（z）≡f（-z）.In general case,we obtained the following result.（2）Let f（z）be an non-constant entire function,β be a real number.if f（z） and f(e^iβz) have two finite IM sharing values,then there exists m∈N₊ such that (e^iβ)^m=1.Secondly,we continue to study the uniqueness of non-constant entire functions with small order,The following result is established.（3）Let f（z） and g（z） be two non-constant entire functions,and satisfyρf<3/4,and let p₁（z） and p₂（z） are two distinct polynomials,if p₁（z） and p₂（z） are IM sharing functions of f（z） and g（z）,and f（z）-g（z） has infinitely many zeros with multiplicity,then f（z）≡ g（z）.On the other hand,in this paper,we discusses the existence of meromorphic function solutions for Fermat-type functional equation with four unknow functions,and the following conclusion is obtained.（4）If non-constant meromorphic function f（z）,g（z）,h（z）,w（z） at most have one simple common pole,and min{ρ_f,ρ_g,ρ_h,ρ_w}<1/2,then the functions of f（z）,g（z）,h（z）,w（z） dose not satisfy the functional equation f¹⁵（z）+ g¹⁵（z）+ h¹⁵（z）+w¹⁵（z）=1.