代数微分方程的值分布与允许解

VALUE DISTRIBUTION AND ADMISSIBLE SOLUTIONS OF ALGEBRAIC DIFFERENTIAL EQUATIONS

研究了非齐次微分多项式与代数微分方程的值分布问题,扩充了仪洪勋的一个结果,作为推论得到了Malmquist型定理。

The following theorems are obtained:Theorem 1. Let f be a non-rational meromorphic function in the plane and let F, H[f], P[f] and Q[f] be differential polynomials in f where neither of them vanishes identically such that H[f] is homogeneous and that F is a polynomial of degree n of f. Then functions of the form ψ=H[F]P[f]+Q[f] satisfy (nd_H-d_Q)T(r,f)≤N(r,1/)+(W_H-d_H+1)N(r,1/F)+(W_Q-d_Q+1)N(r,f)+S(r,f). where d_H and W_H are the degree and weight of the differential polynomial H[f] in frespectively, and so on.Theorem 2. Assumptions as in Theorem 1. If f satisfies the following equationH[F]P[f]=Q[f], then(nd_H-d_Q)T(r,f)≤(W_Q-d_Q) N(r,f)+d_H N(r,1/F)-N(r,1/Q[f])+S(r,f)holds for nd_H≥d_Q and(nd_H-d_Q)T(r,f)≤(W_Q-d_Q) N(r,f)+(W_H-d_H) N(r,1/F)+S(r,f).