涉及微分多项式的亚纯函数的唯一性

UNICITY ON MEROMORPHIC FUNCTIONS WITH THEIRDIFFERENTIAL POLYNOMIALS

本文研究微分多项式具有公共值的亚纯函数的唯一性问题,改进了 Ｉ． Ｌａｈｉｒｉ的有关结果,得到如下结论:设ｆ与ｇ为非常数亚纯函数,ｎ为正整数,ψｎ为ｎ阶常系数线性微分算子．如果(ｉ)f,ｇ以∞为ＩＭ公共值,(ｉｉ)ψｎ(ｆ);ψｎ(ｇ)为非常数且以０,１为ＣＭ公共值,(ｉｉｉ)∑δ(ａ,ｆ)＞１/２,则(ａ)ψｎ(ｆ)ψｎ(ｇ)≡１或者(ｂ)ｆ－ｇ＝ｓ,ｓ是微分方程ψｎ(Ｗ)＝０的解;并且,若(ｉｖ)ｆ至少有一个极点,或者ψｎ(ｆ)至少有一个零点,则结论(ａ)不成立．条件(ｉｉｉ)是精确的．

Abstract In this paper we study the meromorphic functions whose differetial polynomialsshare values, and improve a theorem of Lahiri, and obtain the following results: Let f, g be twononconstant meromorphic functions, n be a positive integer, ψn. is a linear differential operatorof order n with constant coefficients. If (i) f, g share ∞ IM, (ii) ψn.(f), ψn.(g) are nonconstantand share 0, 1 CM, (iii) Σδ(a, f) > 1/2, then either (a) ψn.(f) ψn.(g) ≡ 1, or (b) f - g ≡ sa≠∞.where s is a solution of the differential equation ψn.(W) = 0. Further, if (iv) f has at least onepole or ψn.(f) has at least one zero, then the case (a) does not arise. And the condition (iii) inthis result is sharp.