Uniqueness Of Entire Functions And Difference Polynomials

In 1920s, R.Nevanlinna introduced the characteristic functions of mero-morphic functions and gave the famous Nevanlinna theory [1] which is one of the greatest achievements in mathematics in the 20th century. It plays basic roles for modern researches of meromorphie functions theory [2,3,4,5] and has a very important influence on the development and synretism of many mathematical branches associated to the development of the Nevanlinna theory itself, some new branches have appeared based on main methods and results from Nevanlinna the-ory. Especially, Nevanlinna theory has been used as a very powerful tool and has made the research held more active and popular after it was successfully applied to the research of the global analytic solutions of complex differential equations. Taking advantage of the theory made by himself, R.Nevanlinna studied the conditions with which a meromorphie function can be determined and obtained two celebrated uniqueness theorems on meromorphie functions, which are called Nevanlinna's five-value theorem and four-value theorem. Since then, the research of meromorphie functions began. For over a half century, many abroad and do-mestic mathematicians have devoted themselves to the research and obtained lots of elegant, results on the uniqueness theory. In the past two decades, Professor H. X. Yi did much creative work on the uniqueness theory of meromorphie func-tions, and well improved the development of the uniqueness theory. Moreover, Nevanlinna theory and its extensive has numerous applications in some fields of mathematics, for example, potential theory, complex differential and difference equations, several complex variables, minimal surface, and so on.Difference counterpart of the theory of Nevanlinna theory have been estab-lished very recently. The key result is the difference analogue of the lemma on the logarithmic derivative obtained by Halburd-Korhonen [14] and Chiang-Feng[15], independently. Halburd and Korhonen[16] also established a version of Nevan-linna theory for difference operators. Ishizaki and Yanagihara[17] also developed a version of Wiraan-Valiron theory for slowly growing entire solutions of difference equations. Bergweiler and Langley[18,19] considered the value distributions of difference operators of slowly growing meromorphic functions. Similarly, Nevan-linna theory is also a powerful tool in the research of difference equations. At first, that is the early years in 20th century, the foundation of the theory of complex difference equations was laid by Batchelder [6], Norlund [7], and Whittaker|8] in the early twentieth century. Later on, Shimomura [9]and Yanagihara [10,11.12] studied nonlinear complex difference equations from the viewpoint of Nevanlinna theory. In the year 2007, Halburd and Korhonen [13] proved that, the existence of finite order solutions is a good detector of integrability of difference equations, since then, meromorphic solutions of complex difference equations have become a subject of great interestThe present thesis is a part of the author's work on the value distribution of entire functions and its difference polynomials. It consists of three chapters:In chapter 1, we recall the basic background of Nevanlinna theory and some notations which are always used in our studies. It also includes some classical results in uniqueness theory of value sharing.In chapter 2. we shortly recall difference analogues of the lemma on the logarithmic derivative, of the Clunie lemma, and of the second main theorem and their consequences. Then we makes some research on the entire functions and their difference polynomials, and obtained the following results:Theorem 1 Let f(z) be a transcendental entire function of finite order, and a(z) be a small function with respect to f(z). If n≥2, m∈N. then the difference polynomial f(z)n(f(z)m-1)f(z+c)-a(z) has infinitely many zeros.From the paper-Value distribution of difference polynomials'belonging to I. Lainc and C. C. Yang [20.Theorem 1] and Theorem 1, it is interesting to remark that n can be any positive integer. We also obtained the following result if n=1, under the conditions of Theorem 1: Theorem 2 Under the conditions of Theorem 1.if m(?)3, then the difference polynomial f(z)(f(z)m-1)f(z+c)-a(z)has infinitely many zeros.In Chapter 3,we study the uniqueness of difference polynomials of entire functions sharing one small function,and obtained the following results:Theorem 3 Let f(z) and g(z)be transeendental entire funetions of finite order, and a(z)be a small function with respect to both f(z)and g(z).Suppose that c is a non-zero constant and m,n∈N.if n(?)m+6,and f(z)m-1)f(z-c) and g(z)n(g(z)m-1)g(z+c)share a(z)CM,then we have f(z)≡tg(z),where tm=tn+1=1.Theorem 4 Under the conditions of Theorem 3.If n(?)4m+12,and f(z)n(f(z)m-1)f(z+c)and g(z)n(g(z)m-1)g(z+c) share a(z)IM,then f(z)≡tg(z),where tm=tn+1=1.