单叶函数平均值的叶数

THE VALENCE OF THE CERTAIN MEANS OF UNIVALENT FUNCTIONS

由标准化的单叶函数族中的函数,f(z)和g(z)可以构造新函数F(z)=af(z)+βg(z)和G(z)=z(f(z)/z)~α(g(z)/z)~βα,β∈(0,1),α+β=1.本文讨论了函数F(z)和G(z)在单位圆内的最大叶数,解决了A.W.Goodmam 在1969年提出至今仍未解决,当α,β∈{(0,1)/(1/(1+e(?)))(e(?)/(1+e(?)))}时,F(z)和G(z)在单位圆内的叶数问题.

Given two functions f(z),g(z)in the(usual)class S,the new functions can be formedF(z)=αf(z)+βg(z)and G(z)=z(f(z)/z)~α(g(z)/z)~βwhere α,β∈(0,1)andα+β=1.This paper discusses the valence of the functions F(z)and G(z)and answersthe questions by Goodman in 1969.