一族单叶函数的卷积

Hadamard Products of a Class of Univalent Functions

设■表示D={z:|z|<1}上0的解析函数类,令I(A,B)={f(z)∈■:f'(z)<■,其中-1≤B<A≤1.本文首先建立了I(A,B)中函数的一个从属关系:若f(z)∈I(A,B),则?其中?.本文主要结果是:设f,g∈I(-1,1),则f与g的Hadamard卷积在|z|<γ_0≈0.7336内具有正实部的导数,且γ_0是使得I(A,B)中所有函数的Hadamard乘积在|z|<γ_0内单叶的最大正数.

Let ? denote the class of all analytic functions in the disk D={z: |z|<1}. Let I(A, B) be the class of functions which are in ? and whose derivative is subordinate to (1+Az)/(1+Bz) wnere —1≤B<A≤1. In the first part of this paper, it is shown that for ∫(z) in I(A,B) (f(z))/z is subordinate to (F(z))/z where F(z)= integral from n=0 to z ((1+Az)/(1+Bz))dz. γ_0≈0. 7336 is determinated. The main results of the paper are that derivatives of all Hadamard products in I(—1,1) have positive real part in Dγ_0={z: |z|<γ_0}, and that γ_0 is the largest positive number such that all Hadamard products in I(—1,1) are univalent in Dγ_0.