【摘要】 <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,S₁=S.若 D_f 以原点 w=0 为星形中心,就是说若 w₀∈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更多还原

【Abstract】 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 S₁（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∫₀²πda（θ）=1 satisfying.■Conversely,if the increasing function a（θ）satisfies 1/2π∫₀²πda（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∫₀²πdα（θ）=1Cor 2.Supposing 0≤ρ₁≤ρ₂<1,if f₁（z）S（ρ₁）,then there existsf₂（z）∈S（ρ₂）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≤ρ<l,then for any λ,0≤λ≤1,we have（?）where lim（?）（z）=1.z→0Cor.3.If f（z）∈S,then for any λ,0≤λ≤1,（?）This involves the theorem（?）as a special case^[6].Cor.4.If f（z）∈S,then for any λ,0≤λ≤1,In particular,f’（z）（?）（1-x）²/1.These relations of subordination give the variability region for（?）（z）,i.e.the smallest region containing the values （?）=[（?）（z）]^-1for a fixed z of|z|<1 when f（z）runs over the whole class S（ρ）with 0≤ρ<1.Thisregion is the circle|（?）-1|≤|z|.Thus we obtain the following results;Gor.5,If f（z）∈S,then （?）（0≤λ≤1）.Cor.6.If f（z）∈S,then（?）（0≤λ≤1）.Theorem D.（?）（p=1,2,…）,|z|==r<1,then（i）for 0≤ρ<1,0≤λ≤p,and 2λ（1-ρ）≤p,（?）（ii） for 0≤ρ<1,（?）are sin r^p,（?）These are reduced to equalities when and only when （?）|η|=1.Theorem E.If■≤p<1,then■where 0≤A,B<∞,A²+B²>0.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<l,and 0<γ₁≤γ₂<1,then we have for μ≥0,■■Further,the relations■and■hold true for μ≥1.更多还原