整函数导数的Picard例外集

Picard Sets of Entire Functions and Their Derivatives

主要证明了如下两个结论:(1)设f为超越整函数,且零点重数至少为k+2 (k为正整数),E={λ_n}_(n=1)~∞是复平面中的无限点集,满足|λ_(n+1)/λ_n|>q>1,则f^(k)在C\E中取每个非零有限复数b无穷次。(2)设复数序列a_n和正序列ρ_n满足|a_(n+1)/a_n|>q>1,又设f为超越整函数,且零点重数至少为k+2(k为正整数),则对任何b∈C,b≠0, f(k)-b在U_(n=1)~∞B(a_n,ρ_n)之外有无穷多个零点,其中β>1,B(a_n,ρ_n)={z:|z-a_n|<ρ_N}。

In this paper, it be proved ： （1） Let f is a transcendental entire function, and have only zeros of order at least k＋2, let E={λn}^∞n=1 be an infinite point set in with ｜λn＋1/λn｜〉q〉1. Then f^（k） assumes all values w ∈ C, except possibly zero, infinitely often in C ／ E. （2） Suppose that the complex sequence （αn） and the positive sequence （ρn） satisfy for all n, ｜αn＋1/αn｜〉q〉q,log 1/ρn〉q1/4＋1（k＋3）β（1＋o（1））/q1/4-1 logq（log｜αn｜）^2,（n=1,2,3,...） Then if f is a transcendental entire function, and has only zeros of order at least k＋2, the equation f^（k） = b must have infinitely many solutions outside the union of the discs B（an,pn）, for any b ≠ 0. Here β〉 1 and B（αn,ρn） = {z： ｜z- αn｜ 〈 ρn}.