涉及导数与公共值集的亚纯函数的唯一性

Unicity for Meromorphic Functions Dealing with Derivatives and Sharing Sets

在亚纯函数唯一性理论中，亚纯函数同时涉及导数与公共值集的唯一性问题是一困难而有趣的问题．本文在这方面做了尝试，运用比较简洁的方法，经过细致的计算，把仪洪勋等人的结果由公共值推广到公共值集的情况，得到结果：设ｋ，ｎ为正整数，ｎ≥２，Ｓ１＝｛∞｝，Ｓ２＝｛０｝，Ｓ３＝｛１，ω，ω２，…，ωｎ－１｝，ωｎ＝１为３个集合，若非常数亚纯函数ｆ与ｇ以Ｓ１，Ｓ２为ＣＭ公共值集，ｆ（ｋ）与ｇ（ｋ）以Ｓ３为ＣＭ公共值集，且满足下述２个条件之一：ｉ）ｎ≥５，且δ（０，ｆ）＜１，或Θ（∞，ｆ）＞０；ｉ）２≤ｎ≤４，且２δ（０，ｆ）＋（ｋ＋１）Θ（∞，ｆ）＞ｋ＋２，则ｆ≡ｔｇ，或ｆ（ｋ）·ｇ（ｋ）≡ｔ。

In the uniqueness theory of meromorphic functions, the problem of uniqueness dealing with derivatives and sharing sets is a difficult and interesting one and has been studied in this paper. By using brief methods and detailed calculating, the authors extend from sharing values (Yi H X et al.) to sharing sets, and obtain the following theorem:Let k, n be two positive integers,n2, and S1,S2 and S3 be three sets such thatS1={}, S2={0}, S3={1,,2,,n-1}, n=1.If S1 and S2 are CM sharing sets for nonconstant meromorphic functions f and g; S3 is CM sharing set for f(k) and g(k); and the following conditions holds,i) n5,(0,f)<1, or(,f)>0;ii) 2n4,2(0,f)+(k+1)(,f)>k+2, then ftg, orf(k)g(k)t, where tn=1 .