On a class ofp-valent starlike functions of orderα

Abstract

LetΩdenote the class of functionsw(z),w(0)=0,|w(z)|<1analytic in the unit disc⋃={z:|z|<1}. For arbitrary fixed numbersA,B,−1<A≤1,−1≤B<1and0≤α<p, denote byP(A,B,p,α)the class of functionsp(z)=p+∑n=1∞bnznanalytic in⋃such thatP(z) ϵ P(A,B,p,α)if and only ifP(z)=p+[pB+(A−B)(p−α)]w(z)1+Bw(z),w ϵ Ω,z ϵ ⋃. Moreover, letS(A,B,p,α)denote the class of functionsf(z)=zp+∑n=p+1∞anznanalytic in⋃and satisfying the condition thatf(z) ϵ S(A,B,p,α)if and only ifzf′(z)f(z)=P(z)for someP(z) ϵ P(A,B,p,α)and allzin⋃. In this paper we determine the bounds for|f(z)|and|argf(z)z|inS(A,B,p,α), we investigate the coefficient estimates for functions of the classS(A,B,p,α)and we study some properties of the classS(A,B,p,α).