Sharing Sets And Uniqueness Of Meromorphic Functions With Their Q-differences

The value distribution theory of meromorphic functions established by R.Nevanlinna is also called Nevanlinna theory. Its main contents consist of the two fundamental theorems and the relations between the deficiencies. Besides, it was considered to be one of the most elegant mathematic branches by the famous mathematician WeL. The uniqueness theory of meromorphic functions is the most important part of the Nevanlinna theory. It mainly stud-ies some proper conditions (p) under which there is only one function satisfying (p). It is well known that any polynomial is determined by its zero points except for a non-constant factor. Generally speaking, it is not ture for transcendental entire functions or ordinary meromorphic functions. Nevanlinna proved that the transcendental meromorphic functions can be determined uniquely by five sharing values which was also called the Nevanlinna's five-value theorem. He himself proved the Nevanlinna's four-value theorem, too. So that, it is import to study the uniqueness theory of meromorphic functions carefully. In recent decades, many mathematicians paid close attention to it. With the tool of the value distri-bution established by R.Nevanlinna, the uniqueness theory of meromorphic functions such as five-value theorem, three-value theorem and sharing sets has become an interesting and important field. Many good and new results appear as time goes on. In recent years, the value distribution and uniqueness problems have been extended to the fields of the differ-ence of meromorphic functions, non-Archimedean and multiple complex analysis. Especially in the difference fields. And the new field has attracted many other mathematicians such as W.Bergweiler, R.G.Halburd, Yik-Man Chiang and so on. They also obtained plenty of elegant results on the research of the uniqueness theory.In this paper, we will give some results on the uniqueness of meromorphic functions and their q-differences sharing sets under the guidance of Professor Hu Peichu. It has three chapters.In chapter 1, we briefly introduce the background of this thesis, Nevanlinna basic theory and some fundamental results and notations of the uniqueness which would be used in the next two chapters.In chapter 2, we studied the problems of sharing sets and uniqueness of meromorphic functions with their q-differences and gained the two theorems.Theorem 1. Let Si={1, w,…,wm-1},S2={∞}, where and Let q is a non-zero complex constant. Suppose that f(z) is a nonconstant meromorphic function of zero order such that Ef(z)(Sj)=Ef(qz)(Sj) (j=1,2). Ifn≥4, then f(z)=±tf(qz), where tn=1.Let f(z) be an entire function, the condition n≥4 can be reduced to n≥3.Corollary 1. Let S={1,w,…,wn-1}, where and Let q is a non-zero complex constant. Suppose that f(z) is a nonconstant entire function of zero order such that Ef(z)(S)= Ef(qz)(S). If n≥3, then f(z)=±tf(qz), where tn=1.If we choose a proper set S1, we can gain the exact function f(qz).Theorem 2. Let m≥2,n≥2m+4 with n and n - m having no common factors. Let a and b be two non-zero constants such that the equation wn+awn-m+b= 0 has no multiple roots. Let S1={w|wn+awn-m+b=0},S2={∞}, and q is a non-zero complex constant. Suppose that f(z) is a nonconstant meromorphic function of zero order. Then Ef(z)(Sj)=Ef(qz)(Sj) (j=1,2) imply f(z)=f(qz).We have the following result concerning entire function.Corollary 2. Let n≥5 be an integer and let a and b be two non-zero constants such that the equation wn+awn-m+b=0 has no multiple roots. Let S={w|wn+awn-m+b=0}, and q is a non-zero complex constant. Suppose that f(z) is a nonconstant entire function of zero order. Then Ef(z)(S)= Ef(qz)(S) implies f(z)= f(qz).In chapter 3, we studied the problems of sharing the small function and uniqueness of meromorphic functions with their q-difference polynomials. Main results are stated as follows.Theorem 3. Let f(z) be a transcendental meromorphic function of zero order, and a(z) be a small function with respect to f(z). Suppose that where qj(j=1,2,…,n) are distinct values in C and uj(j=1,2,…,n) are positive integers. Let , and at least one of the uj≥2. If the exponent of convergence of the poles of f is zero, then F(z)-α(z) has infinitely many zeros, and on a set of logarithmic density 1.Considering of the difference polynomial f(z)n(f(z)-1)f(qz)-α(z), we getTheorem 4. Let f (z) be a transcendental entire function of zero order, and a(z) be a small function with respect to f(z). Suppose that q is a non-zero complex constant and n is an integer. If n≥2, then f(z)n(f(z)-1)f(qz)-a(z) has infinitely many zeros on a set of logarithmic density 1.Theorem 5. Let f(z) and g(z) be two transcendental entire functions of zero order, andα(z) be a small function with respect to both f(z) and g(z). Suppose that q is a non-zero complex constant and n(> 7) is an integer. If f(z)n(f(z)-1)f(qz) and g(z)n(g(z)-1)g(qz) shareα(z) CM, then f(z)=g(z) on a set of logarithmic density 1.