Functional equations related to family

By Nevanlinna's value distribution theory, we give a more precise form of the meromorphic function f provided that f m can be expressed as a summation of n + 1 (n!≤! m) functions in family The results in this article generalize some known results obtained by Green, M., 1974, On the functional equation a new Picard theorem. Transactions of the American Mathematical Society, 195, 223–230; Noda, Y., 1993, On the fuctional equation and rigidity theorems for holomorphic curves. Kodai Mathematical Journal, 16, 90–117.