Uniqueness Theory On Entire Functions And Their Linear Differential Polynomials

In 1920s, R.Nevanlinna introduced the characteristic functions of meromorphirfunctions, proved two fundamental theorems.and established so-called value distribute theory or called Nevanlinna theory. This theory is one of the most importantachievements in the 20th century praised by H.weyl. It plays basic roles for modern researches of meromorphic function theory and has a very important inflnence on the development and syncretism of many mathematical branches. Associatedto the development of the Nevanlinna theory itself, some new mathematical branches have appeared based on main methods and results from Nevanlinna theory.In the late 1920s, R.Nevanlinna also studied the conditions which determine completely a meromorphic function, and obtained three celebrated uniqueness theoremsfor meromorphic functions, which are usually called Nevaulinna's five-value thenrcm, four-value theorem and threc-valuc theorem respectively. This launched the investigation of uniqueness theory of meromorphic functions.The present thesis is a part of the author's research work on the uniqueness problems of entire functions that share values or small functions with their linear differential polynomials over C or Cⁿ. It consists of three chapters.In chapter 1. we briefly introduce the background of this thesis, which containssome fundamental results and notations of Nevanlinna theory. In chapter 2, we improve the results of Bernstcin-Chang-Li and Li-Yang by studying the uniqueness problem of entire functions that share two values with their linear differential polynomials. The main result is the following:Theorem 1. Let f be a non-constant entire function, and let L(f) be a linear differential polynomial in f of the following formwhore the coefficients b_n((?)0), b_n-1,…., b₀, b_-1 are small meromorphic functions of f. Assume that f and L(f) share two distinct finite values a₁, a₂ IM, and all the simple a₁-points of L(f) are simple a₁-points of f. Then f and L(f) share the value a₁ CM, and So either f≡L(f), or they have the following expressionsandwhereαis a non-constant entire function.Corollary 2. Under the conditions of Theorem 1. if we furthermore assume that f and L(f) share two distinct finite valuesα₁,α₂ IM, and all the simpleα₁-points andα₂-points of L(J) are simpleα₁-points andα₂-points of f respectively, then f and L(f) share the valuesα₁.α₂ CM and f≡L(f).In Chapter 3, we makes the research of uniqueness problem of entire functionson Cⁿ that share one small function with their linear differential polynomials. The main result is the following: Theorem 3. Let f be a transcendental entire function on Cⁿ, let L(f) be a linear differential polynomial in f of the following form:where a(l₁,l₂,…l_n)∈C are constants and at least one of a(l₁,l₂,…l_n)≠0 (l₁+ l₂+…+ l_n = k), and let a(z) be a small meromorphic function of f such that a(z)(?) 0,∞. If f and L(f) share a(z) CM, andδ(0,f) > 1/2, then f(?) L(f).