涉及fⁿf^(k)+H(f)-b 的零点重级的正规定则

A Normality Criterion Concerning the Zeros’ Multiplicity of fⁿf^(k)+H(f)-b

摘要: 本文研究全纯函数族的正规性,证明了如下结论：设M，n，k为三个正整数，其中当n=k=1 时，M≥9 ；当nk>1 时， ，b 为一个非零有穷复数，设F 为区域D 内的一族全纯函数，H(f) 为f 的微分多项式且满足 ，若对于F 中的每一个函数f(z) 均有(1)f(z) 的零点重级≥k ；(2)f⁽ⁿ⁾f^(k)+H(f)-b 的零点重级≥M ，则F 在D 内正规。

Abstract: In this paper, we study the normality of holomorphic functions and prove the following results: Let M, n, k be three positive integers satisfying M≥9 when n=k=1 and when nk>1, b(≠0) , is a finite complex number; let F be a family of holomorphic functions in a domain D and H(f) be a differential polynomial of f and satisfy , if for each f∈ F  , satisfies (1) all zeros of f have multiplicity at least k; (2) all zeros of f⁽ⁿ⁾f^(k)+H(f)-b have multiplicity ≥M ,  then F  is normal in D  .