某些单叶函数的积分

INTEGRALS OF CERTAIN UNIVALENT FUNCTIONS

设K(μ)(0≤μ<1)和S(m,M)(|m-1|<M≤m)分别是E={z:|z|<1}内满足条件Re{1+zf″(z)/f′(z)}>μ和|zf′(z)/f(z)-m|<M的解析函数f(z)=z+a_2z^2+…+a_nz^n+…构成的类。本文的目的是研究形如 integral from n=0 to z {f′(t)}~α{g(t)/t}~βdt的积分,这里0<α≤1,β是实(或复)常数,f(z)∈K(μ),g(z)∈S(m,M)。我们的结果纠正了Pandey和Bhargava[1]的一个主要结果。

Let K(μ), 0≤μ≤1, and S(m, M), |m-1|<M<m, be the classes of functions f(z)=z+a_2z^2+…+a_nz^n+…regular and satisfying the conditions Re{1+zf~″(z)/f′(z)}>μand |zf~′(z)/f(z)-m|<M in |z|<1 respectively. The object of this note is to study the integrals integral from n=0 to z {f′(t)}~α{g(t)/t}~βdt, whereβ is a real (or complex) constant, 0<α≤1, f(z)∈K(μ) and g(z)∈S(m, M). Our result corrects a result of Pandey and Bhargava [1].