关于整函数与亚纯函数唯一性定理的推广

GENERALIZATION OF UNICITY THEOREMS FOR ENTIRE OR MEROMORPHIC FUNCTIONS

讨论了整函数与亚纯函数的唯一性问题,改进了Ueda和本文第一位作者的有关定理。

If two meromorphic functions f(z) and g(z) have the same a-points with the same multiplicities, we denote this by f=a■g=a.The following theorems are proved: Theorem l. Let a_l,a_2, …,a_p be p distinct finite complex numbers, and a_i≠0, 1(i=1,2,…,p). Suppose that f(z) and g(z) are nonconstant meromorphic functions, and f(z)■g(z). If f=0■g=0, f=1■g=1, f=∞■g=∞,and sum from i=1 to p (δ(a_i,f)>2p/p+3)then there exists one a_j in a_1,_a_2, …,a_p such that a_j is a Picard exceptional value of f(z), and f(z) and g(z) must satisfy exactly one of the following relations:1) (f-a_j)(g+a_j-1)≡a_j,(1-a_j),2) f-(1-a_j,)g≡a_j,3) f≡a_jg.Theorem 2. Let a_1, a_2,…,a_p be p distinct finite complex numbers, and a_i≠0, 1(i=1,2,…, p). Suppose that f(z) and g(z) are nonconstant entire functions, and f(z)≡g z). If f=0■g=0, f=1■g=1,and sum from i=1 to p(a_i,f)>p/(p+2),then there exists one a_j in a_1,a_2,…,a_p such that a_j and 1-a_j are Picard xceptional value of f(z) and g(z) respectively, and (f-a_j)(g+a_j-1)≡a_j(1-a_j).