亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

文4]对简单形式的微分多项式fkf’+a的零点分布进行了讨论,文1]对一般形式的微分多项式fkQf]+Pf]的零点分布进行了讨论.但由于极点给证明带来的困难,这些工作主要是对整函数来做的.本文证明了任一满足δ(∞,f)>k+2ΓQ+3ΓP+2/2k+2ΓQ+1的超越亚纯函数f,微分多项式fkQf]+Pf]在不含f,Qf]极点和Pf]零、极点的可数个圆盘并集之外有无穷多个零点,其中k≥3Γp+2,而ΓQ,ΓP分别是f的微分多项式Qf],Pf]的权.文1]和2,4,6]中的结论是本文结论的特殊情况.

The Distribution of Zeros for Differential Polynomials f~kQ[f] + P[f]

The distribution of zeros for differential polynomials fkf'+ a and fkQf] + Pf] has been studied in 4] and 1]. There are some difficulties in proving zeros distribution of differential polynomials because of the presence of poles. So the above results are only for entire functions. In this paper, we prove that for a transcendental meromorphic function f and two differential polynomials Qf], Pf] of f, fkQf] + Pf] has infinitely many zeros outside the union of the disce which do not contain the poles of f,Qf], Pf] and the zeros of Pf]. Where f satisfies δ(∞, f) >and P are weights of Qf] and Pf]. The result improves that of 1] and 2,4,6].