TWO MEROMORPHIC FUNCTIONS SHARING SETS CONCERNING SMALL FUNCTIONS

Abstract

The main purpose of this paper is to deal with the unique- ness of meromorphic functions sharing sets concerning small functions. We obtain two main theorems which improve and extend strongly some results due to R. Nevanlinna, Li-Qiao, Yao, Yi, Thai-Tan, and Cao-Yi. It is well known that two nonconstant polynomials f and g over an algebraic closed field of characteristic zero are identical if there exist two distinct values a and b such that f(x) = a if and only if g(x) = a and f(x) = b if and only if g(x) = b. In 1926, R. Nevanlinna (3) proved his famous five-value theorem that for two nonconstant meromorphic functions f and g on the whole complex plane C, if they have the same inverse images (ignoring multiplicities) for five distinct values, then f(z) · g(z). After this very work, the uniqueness of meromorphic functions with shared values onC attracted many investigations (for references, see (10)). It is very interesting to consider distinct small functions instead of distinct complex numbers on C. Over the last few years, there were several generaliza- tions of Nevanlinna's five-value theorem. To state some of these results, we must introduce some notions. Let h be a nonzero holomorphic function on C, expanding h as h(z) = P1 i=0 bi(ziz0) i around z0, then we define h(z0) := min{i : bi 6 0}. Let k and M be positive integers or +1. We set M h (z) = min{M,h(z)},