Radius of Starlikeness for Class S(a,n)and Its Extension

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)-21-α-β-n(1-α)]r~+1=0.0≤α≤α_0,(1-α)1-(1-2β)r~]-nr~(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 thecxtension of well-known resultsT.Th3] and 8,Th3]

Radius of Starlikeness for Class S(a,n)and Its Extension