关于线性微分多项式的值分布

On value distribution of linear differential polynomials

下述定理得到证明 :设 f是一超越亚纯函数 ,ψ[f]是 f的一个k阶线性微分多项式 .如果微分方程 (ψ[ω])′ =0的亚纯解ω均为 (ψ[f])′的小函数 ,则对任意正数ε及任意 q(q≥ 2 )个判别有穷复数bj(j =1,2 ,… ,q) ,恒有(q - 1) (1- 1kq +q- 1)T(r ,ψ[f]) < qj =1N(r ,1ψ[f]-bj) +εT(r ,ψ[f]) +S(r ,ψ[f])且对一切有穷复数b ,有∑b≠∞δ(b ,ψ[f]) ≤ 12k+1+1上述结果是杨乐的两个定理的推广 .定理中的条件“(ψ[ω])′ =0的亚纯解ω均为 (ψ[f])′的小函数”是必要的 .

The following theorem is proved:Given f is a transcendental meromorphic function,and a 0,a 1,…,a k(a k0)are some small functions of defined by ψ[f]=a 0f+a 1f′+…+a kf (k) .If allω with(ψ[ω])′=0 are the small functions of (ψ[f])′,then for ε>0 and b j∈C(j=1,2,…,q),there exists (q-1)(1-1kq+q-1)T(r,ψ[f])<qj=1N(r,1ψ[f]-b j)+εT(r,ψ[f])+S(r,ψ[f])and for all b∈C,there existsb≠∞δ(b,ψ[f])≤12k+1+1Above mentioned two inequalities includes Yang Lo's two inequalities.