In this paper, we study the uniqueness problem on difference polynomials and its differential of meromorphic function sharing a common value. | UNIQUENESS OF MEROMORPHIC FUNCTION AND ITS ORDER K CONCERNING THE DIFFERENCE POLYNOMIALS PHAM TUYET MAI Abstract. In this paper, we study the uniqueness problem on difference polynomials and its differential of meromorphic function sharing a common value. 1. Introduction A meromorphic function means meromorphic in the whole complex plane. We assume that the reader is used to doing the standard notations and fundamental results of Nevanlinna theory. Let be two meromorphic function f, g and a ∈ C ∪ {∞}. We say that f and g share a − CM if f − a and g − a have the same zero with multiplicities . We denote by Em) (a; f ) = {z ∈ C : f (z) = a} the set of all a-points of f with multiplicities not exceeding m, where a-point is counted according to it’s multiplicity. In 2011, K. Liu, X. Ling and T. B. Cao proved the following: Theorem A. Let f and g be transcendental meromorphic functions with finite order, c ∈ C be a nonzero constant and n ∈ N. If n > 14, f n (z)f (z + c) and g n (z)g(z + c) share 1 − CM , then f = tg, or f g = t, where tn+1 = 1. The results of this paper was suggested thinking of ideal differential order k. We will consider the functions (f n (z)f (z + c))(k) and (g n (z)g(z + c))(k) . Our result is stated as follows: Theorem 1. Let f and g be transcendental meromorphic functions with finite order, c ∈ C is a nonzero constant and n ∈ N, k be a positive integer. If one of the following conditions is hold 1. n > 10k + 24 and E1) (1, (f n (z)f (z + c))(k) ) = E1) (1, (g n (z)g(z + c))(k) ); 2. n > 4k + 15 and (f n (z)f (z + c))(k) , (g n (z)g(z + c))(k) share 1 − CM ; then f = tg or (f n (z)f (z + c))(k) .(g n (z)g(z + c))(k) = 1, where tn+1 = 1. 2000 Mathematics Subject Classification. Primary 32H30. Key words: Uniqueness theorem, difference polynomials. 1 142Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên 2 Theorem 1 gives some uniqueness results for transcendental meromorphic functions with finite order and it’s .
