Frank-Weissenborn不等式的推广

A generalization of Frank Weissenborns inequality

下述定理得到证明：设ｆ是超越亚纯函数，ａ０，ａ１，…，ａｋ是ｆ的一组小函数，且ａｋ≠０．置Ｄ［ｆ］＝ａ０ｆ＋ａ１ｆ′＋…＋ａｋｆ（ｋ）如果微分方程Ｄ［ω］＝０的亚纯解ω均为ｆ的小函数，则对任意的正数ε，都有（ｋ－１－ε）Ｎ（ｒ，ｆ）＜Ｎｒ，１Ｄ［ｆ］＋（１＋ε）Ｎ１（ｒ，ｆ）＋Ｓ（ｒ，ｆ）此不等式使著名的ＦｒａｎｋＷｅｉｓｓｅｎｂｏｒｎ不等式成为其特殊情况．

In this paper,the following theorem is shown: let f be a transendental meromorphic function and a 0,a 1,…, a k (a k≠0) be some small functions of f and define by D=a 0f+a 1f′+…+a kf (k) If all ω s with D =0 are the small functions of f,then for every ε >0, (k-1-ε)(r,f)<Nr,1D+(1+ε)N 1(r,f)+S(r,f) this result includes Frank Weissenborns inequality.