劳勃生的特殊星像函数和特殊凸像函数

<正> 设函数w在单位圆 E_z:|z|<1上是正则的.假如f(z)在 E_z上是单叶的,那末 D_f=f(E_z)是 w 平面上单叶的区域.记这种单叶函数f(z)的全体为 S_p,S_1=S.若 D_f 以原点 w=0 为星形中心,就是说若 w_0∈D_f则缐段■整个地落在区域 D_f 中,称这种函数 f(z)是 E_z 中的星像函数,其特徵是在 E_z

SOME CLASSES OF FUNCTIONS OF STAR-LIKENESS

In the present paper,we consider the following classes of functions:S_p(ρ):the class of functions f(z)=■regular and schlichtin the unit circle |z|<1,and being such thatR(zf’(z)/f(z))≥ρ,0≤ρ<1,|z|<1.For simplicity,we write S_1(ρ)=S(ρ),S(0)=S,S(1/2)=S.K(ρ):the sub-class of S,whereof each function f(z)be such thatzf'(z)∈S(ρ),so that(?)|z|<1.Every function f(z)of S(ρ)(0≤ρ<1)is star-like with respect tothe origin f(0)=0,in particular,any function w=f(z)of K(ρ)maps|z|<1 onto a convex domain in the w-plane.Our main results are as follows:Theorem A.Corresponding to α function f(z)of S(ρ),0≤ρ<1,there exists an increasing function a(θ)with 1/2π∫_0~2πda(θ)=1 satisfying.■Conversely,if the increasing function a(θ)satisfies 1/2π∫_0~2πda(θ)=1,thenthis formula of representation implies f(z)∈S(ρ). Cor.1.A necessary and sufficient condition for f(z)∈(ρ) is thatf(z)can be written as■with some increasing function α(θ)satisfying 1/2π∫_0~2πdα(θ)=1Cor 2.Supposing 0≤ρ_1≤ρ_2<1,if f_1(z)S(ρ_1),then there existsf_2(z)∈S(ρ_2)satisfying(?)In particular~(3]),f(z)∈S implies 2(?).(?)Theorem B.If f(z)=(?),0≤ρ<1,then(?)Theorem C.If f(z)∈S(ρ),o≤ρ0.The signs of equality can hold whenand only when f(z)=z(1-ηz~p)~((-2/p)(1-ρ)),|η|=1.In particular,if f(z)∈S,then■and if f(z)∈S,then■Theorem F.Let p be an odd integer.■ 0≤ρ