Value Distribution And Uniqueness Of Difference Polynomials Of Meromorphic Function

The theory of value distribution and uniqueness of meromorphic functions is an important subject in complex analysis. Many researchers have made a great number of contribution to it and obtained a lot of important results. In this paper, we mainly study the value distribution and uniqueness of difference polynomials of meromorphic functions. This article is divided into four parts.In part one, we introduce the background of research for the theory of value distribution and uniqueness of difference polynomials, then give the existing results and main results of this article.In part two, the elementary knowledge and simple marks of the Nevanlinna theory as well as the fundamental definitions and results of the value distribution and uniqueness of meromorphic functions have been introduced.In part three, we study the value distribution of q-shift difference polynomials of meromorphic function with zero order by using Nevanlinna theory and obtain the following results mainly. The results improve the condition n 37 of Theorem D and n 38 of Theorem E.Theorem 3.1.1 Let f be a transcendental meromorphic function with zero order, m be a non-negative integer, n be a positive integer, a, q ?\ {0}. If n 35, then()(())[()()]()n mf z f z -a f qz +c -f z -az has infinitely many zeros, where a)(z is a nonzero small function with respect to f.Theorem 3.1.2 Let f be a transcendental meromorphic function with zero order and a, q be nonzero complex constants. Then, for n 35,() [()()]nf z +a f qz +c -f z assumes every value b ? infinitely often.In part four, we study the value distribution and uniqueness of difference polynomials by using Nevanlinna theory of meromorphic functions, improving the previously known results.First, we conside the value distribution of difference polynomials and prove the following two theorems mainly.Theorem 4.1.1 Let f be a transcendental meromorphic function with zero order, a)(z is a nonzero small function with respect to f, q ?\ {0}. If n 34, then()()()nf z f qz +c -az has infinitely many zeros. The conclusion is also true for any finite-order transcendental meromorphic function when q =1.Theorem 4.1.1 improves the condition n 36 in Theorem A1 to n 34, Example 4.1.1 below shows that the condition in Theorem 4.1.1 can not be improved.Theorem 4.1.2 Let f be a transcendental meromorphic function with zero order, a)(z is a nonzero small function with respect to f, a, q ?\ {0}. If n 34, then()()()nf z +af qz +c -az has infinitely many zeros. The conclusion is also true for finite-order transcendental meromorphic function when q =1, The following Example 4.1. 2 illustrates the condition in Theorem 4.1.2 is the best.Then, we conside the uniqueness of difference polynomials and prove the following three theorems mainly.Theorem 4.1.5 Let f, g be two transcendental meromorphic functions with zero order, q ?\ {0} and n 36. If()()nf z f qz and()()ng z g qz share 1, ￥ CM, then 1()(), 1nf z tg z t+o =.Theorem 4.1.5 improves the condition n 38 in Theorem D1 to n 36.Theorem 4.1.6 Let f, g be transcendental meromorphic functions with zero order, q ?\ {0} and n 35. If()(() 1)()nf z f z -f qz and()(() 1)()ng z g z -g qz share 1 CM, then f(z) og(z).Theorem 4.1.6 improves the condition n 36 in Theorem E1 to n 35.Theorem 4.1.7 Let f, g be transcendental entire functions with zero order, m, n+?, and n 3m +4. If()(())()n mf z f z -a f qz +c and()(())()n mg z g z -a g qz +c share P(z)( o/0) CM, then f(z) og(z).Theorem 4.1.7 improves the condition n 3m +5 in Theorem F1 to n 3m +4.