涉及微分多项式的亚纯函数正规性

Normality of meromorphic functions with differential polynomial

研究了涉及微分多项式的亚纯函数的正规性.继承Schwick的思想将正规族与分担值联系起来,对一族亚纯函数中函数与该函数微分多项式分担值的情况进行研究,得出亚纯函数的正规性.已知定理:设F为区域D上的全纯函数族,k为正整数,a,b,c和d为有穷复数,b≠0,c≠0且b≠a,若对f∈F,f-d的零点重级至少为k,f=0■f(k)=a且f(k)=b■f=c.则F在D上正规.本文将这个定理推广到亚纯函数情形,并且将f(k)用f的微分多项式来代替,结论仍成立.

The normality of the linear combinations, whose coefficients are analytic functions on D,of the derivatives of a meromorphic function on D was studied. The main idea is to extend Schwick＇s ideas to make the normal family link to the shared values, by studying the function from a family of meromorphic functions, that share values with a differential polynomial of this function, to abtain the normality of the meromorphic functions. It already has proved：Let F be a family of holomorphic functions on a domain D, k a positive integer, a, b, c and d finite complex values with ,b≠0,c≠0, b≠a. If for each f ∈F,all the zeros of f- d are of multiplicity at least k ,such that f = 0=〉f^（k） = a and f^（k） = b=〉f = c, then F is normal on the domain D. By using an analogue to above theorem for meromorphic functions, and replacing the function f^（k） with a differential polynomial of f, a normal criterion was proved.