Research On The Uniqueness Of Meromorphic Functions And Their Difference Operators

Since the famous Finnish mathematician R.Nevanlinna founded the Nevanlinna theory in 1925,its theory has been widely used in various branches and applications of complex analysis.R.Nevanlinna obtained the famous five-value theorem and four-value theorem of meromorphic functions by using the relevant results of Nevanlinna theory,which laid a foundation for the development and research of the uniqueness theory of meromorphic functions.In this paper,we discuss the uniqueness problem of meromorphic functions and their difference operators.The uniqueness problems of f’(z)and f(z+c),f(z)andΔf(z)are transformed into complex differential difference equations.Some results are obtained,which further enrich the uniqueness theory of meromorphic functions.The general research framework and research results of this paper are as follows:In the first chapter,this chapter mainly introduces the development history and research status of the uniqueness theory of meromorphic functions.In the second chapter,this chapter mainly introduces the basic definitions and related symbols in the uniqueness theory of meromorphic functions and the main conclusions.In the third chapter,this chapter mainly studies the uniqueness problem when f’(z)and f(z+c)share two common values.We propose the problem to be studied from the existing theorem A.Theorem A[36]Let f(z)be a transcendental entire function of finite order,and let a(≠0)∈ C.If f’(z)and f(z+c)share 0,a CM,then f’(z)≡f(z+c).Here,we pose a list of questions related to Theorem A.1.If the condition " f’(z)and f(z+c)share 0,a CM" is changed to " f’(z)and f(z+c)share two distinct values a,b CM," is Theorem A still true?2.Can value sharing condition or the restriction on the order of f(z)be improved in Theorem A?Thus,the theorems to be explored in this chapter are obtained,namely Theorem 3.1.1 and Theorem 3.1.2.Theorem 3.1.1.Let f(z)be a transcendental entire function of hyper-order strictly less than 1.If f’(z)and f(z+c)share two distinct values a,b CM,then f’(z)≡f(z+c).Theorem 3.1.2.Let f(z)be a transcendental entire function of hyper-order strictly less than 1,and let a,b be two distinct constants.If f(z)and Δf(z)=(f(z+c)-f(z))((?)0)share a,b CM,then f(z)≡ Δf(z).In the fourth chapter,this chapter mainly studies the uniqueness problem when f’(z)and f(z+c)share a common value.In the existing research results,some results have been given for the " 1CM+1IM" and"2CM" and " 2IM" cases.In this chapter,we will give the case where f’(z)and f(z+c)share a non-zero value a CM or IM.Theorem 4.1.1.Let f(z)be a transcendental entire function of hyper-order strictly less than 1,and let a(≠0)∈C.If f’(z)and f(z+c)share a CM and δ(0,f)>1/2.Then f’(z)≡f(z+c).For the sharing assumption "1 IM",we obtainTheorem 4.1.2.Let f(z)be a transcendental entire function of hyper-order strictly less than 1,and let a(≠0)∈ C.If f’(z)and f(z+c)share a IM and δ(0,f)>4/5.Then f’(z)≡f(z+c).The fifth chapter,this chapter is the summary and prospect of the above research results.