On a class of p-valent starlike functions of order α

Abstract

Let Ω denote the class of functions w(z), w(0)=0, |w(z)|<1 analytic in the unit disc ⋃={z:|z|<1}. For arbitrary fixed numbers A, B, −1<A≤1, −1≤B<1 and 0≤α<p, denote by P(A,B,p,α) the class of functions p(z)=p+∑n=1∞bnzn analytic in ⋃ such that P(z) ϵ P(A,B,p,α) if and only if P(z)=p+[pB+(A−B)(p−α)]w(z)1+Bw(z), w ϵ Ω, z ϵ ⋃. Moreover, let S(A,B,p,α) denote the class of functions f(z)=zp+∑n=p+1∞anzn analytic in ⋃ and satisfying the condition that f(z) ϵ S(A,B,p,α) if and only if zf′(z)f(z)=P(z) for some P(z) ϵ P(A,B,p,α) and all z in ⋃. In this paper we determine the bounds for |f(z)| and |argf(z)z| in S(A,B,p,α), we investigate the coefficient estimates for functions of the class S(A,B,p,α) and we study some properties of the class S(A,B,p,α).