We introduce notations E(r)={z:|z|>r},E=E(1), and suppose It is considered in [2] that F(z) is a starlike function in E (2^(1/2)), if The purpose of this paper is to give adifferent proof of this theorem, and exten...We introduce notations E(r)={z:|z|>r},E=E(1), and suppose It is considered in [2] that F(z) is a starlike function in E (2^(1/2)), if The purpose of this paper is to give adifferent proof of this theorem, and extend this theorem as follows: Suppose Let ∑(λ) denote the class of all functions (1) satisfying the condition where d_n(λ) is the coeffients of the development Then the starlike radius r_0 of the class ∑(λ) is the unique solution of the algebric equation λr^(2(k+1))-2λr^(2(k+1))+λr^(2k)-(k+1-λ)r^2+k-λ=0 in (1,∞).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We introduce notations E(r)={z:|z|>r},E=E(1), and suppose It is considered in [2] that F(z) is a starlike function in E (2^(1/2)), if The purpose of this paper is to give adifferent proof of this theorem, and extend this theorem as follows: Suppose Let ∑(λ) denote the class of all functions (1) satisfying the condition where d_n(λ) is the coeffients of the development Then the starlike radius r_0 of the class ∑(λ) is the unique solution of the algebric equation λr^(2(k+1))-2λr^(2(k+1))+λr^(2k)-(k+1-λ)r^2+k-λ=0 in (1,∞).
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]
