Meromorphic Functions On The Two Issues

In this paper, we consider two problems in the theory of meromorphic functions. First, we study a problem which belongs to T.W.NG in document[19] and get:Theoreml Let f be a transcendental entire function such that f' has at least two distinct zeros .Let g be a nonlinear entire function which permutes with f. Then there exists a ∈ C and m₀ 6 N, g'(a) = 0 such that f^m0 - a and (f^m0)'o g have infinitely many distinct common zeros.Theorem2 Let f be a transcendental entire function such that f' has at least two distinct zeros .Let g be a nonlinear entire function which permutes with f , andthere exists n ∈ N (n > 1) satisfies ,(E (?) R⁺, mesE < +∞).Then there exists a ∈ C , g'(a) = 0 such that f — a and f'o g have infinitely many distinct common zeros.Second ,we study the normality criteria for families of holomorphic functions and get:Theorem3 Let F be a family of holomorphic functions on the unit disk Δ,and let a(≠ b),b(≠ 0),c(≠ 0) be three finite complex numbers. If E_{f( 1)}(0) = E_f(a),(E_f|—)(b)(?)(E_f(-1)|—)(c) for every f∈F, where f^-1(z) =∫_z0^z f(ξ)dξ,(z₀,z ∈ Δ), then F is normal on Δ.Corollary Let F be a family of holomorphic functions on the unit disk Δ,and let a,b(≠ 0) be two distinct finite complex numbers. If E_f(-1)(a) = E_f(a),(E_f|—)(b)(?)E_f(-1)(b) for every f ∈ F, then F is normal on Δ.