设f（z）=z+sum from v=1 to∞（avzv）是单位圆|z|<1内的解析函数,用N记这种函数的全体.MacGregor研究了N中函数f（z）的单叶星象性,得到若干结果.本文推广了这些结果.1.概念与记号设fp（z）=z+sum from k=1 to∞(akp+1zkp+1)是|z|&...设f（z）=z+sum from v=1 to∞（avzv）是单位圆|z|<1内的解析函数,用N记这种函数的全体.MacGregor研究了N中函数f（z）的单叶星象性,得到若干结果.本文推广了这些结果.1.概念与记号设fp（z）=z+sum from k=1 to∞(akp+1zkp+1)是|z|<1内的p次对称单叶解析函数,其全体记为SP（P=1,2,…）.特别简记S1=S.如果fz∈Sp,且有β∈[0,1)使得Re{zf′p（z）/fp（z）}>β（|z|<l）,则称fp（z）为P次对称β级星象函数,其全体记为Sp（β）,特别简记:Sp（0）=Sp,SP（1/2）=Sp。展开更多
This paper obtain that the radius of starlikeness for class S(α,n)in [1] is,tespectivety, where α_ is unique solution of equation (αα)^(1/2)=σwith a in (0.1),and α-[1+(1-2α)r^(2n)]/(1-r^(2n)),σ =[1-(1-2α)r~]...This paper obtain that the radius of starlikeness for class S(α,n)in [1] is,tespectivety, where α_ is unique solution of equation (αα)^(1/2)=σwith a in (0.1),and α-[1+(1-2α)r^(2n)]/(1-r^(2n)),σ =[1-(1-2α)r~]/(1+r~).Futhermore,we consider an extension of class S(α,n):Let S(α、β、n) denote the class of functions f(z)=z+α_z^(n+1)+…(n≥1)that are analytie in |z|<1 such that f(z)/g (z)∈p(α,n)[1],where g(z)∈S~*(β)[2].This paper prove that the radius of starlikeness of class S(α, β,n) is given by the smallest positive root(less than 1)of the following equations (1-2α)(1-2β)r^(2)-2[1-α-β-n(1-α)]r^+1=0.0≤α≤α_0, (1-α)[1-(1-2β)r~]-n[r^(1+r^)=0.,α_0≤α<1. where α=[1+(1-2α)r^(2)]/(1-r^(2)(0≤r<1),α_0(?(0,1) is some fixed number.This result is also the cxtension of well-known results[T.Th3] and [8,Th3]展开更多
文摘设f（z）=z+sum from v=1 to∞（avzv）是单位圆|z|<1内的解析函数,用N记这种函数的全体.MacGregor研究了N中函数f（z）的单叶星象性,得到若干结果.本文推广了这些结果.1.概念与记号设fp（z）=z+sum from k=1 to∞(akp+1zkp+1)是|z|<1内的p次对称单叶解析函数,其全体记为SP（P=1,2,…）.特别简记S1=S.如果fz∈Sp,且有β∈[0,1)使得Re{zf′p（z）/fp（z）}>β（|z|<l）,则称fp（z）为P次对称β级星象函数,其全体记为Sp（β）,特别简记:Sp（0）=Sp,SP（1/2）=Sp。
文摘This paper obtain that the radius of starlikeness for class S(α,n)in [1] is,tespectivety, where α_ is unique solution of equation (αα)^(1/2)=σwith a in (0.1),and α-[1+(1-2α)r^(2n)]/(1-r^(2n)),σ =[1-(1-2α)r~]/(1+r~).Futhermore,we consider an extension of class S(α,n):Let S(α、β、n) denote the class of functions f(z)=z+α_z^(n+1)+…(n≥1)that are analytie in |z|<1 such that f(z)/g (z)∈p(α,n)[1],where g(z)∈S~*(β)[2].This paper prove that the radius of starlikeness of class S(α, β,n) is given by the smallest positive root(less than 1)of the following equations (1-2α)(1-2β)r^(2)-2[1-α-β-n(1-α)]r^+1=0.0≤α≤α_0, (1-α)[1-(1-2β)r~]-n[r^(1+r^)=0.,α_0≤α<1. where α=[1+(1-2α)r^(2)]/(1-r^(2)(0≤r<1),α_0(?(0,1) is some fixed number.This result is also the cxtension of well-known results[T.Th3] and [8,Th3]
