On a class ofp-valent close-to-convex functions of orderβand typeα

Abstract

LetS(A,B,p,α)denote the class of functionsg(z)=zp+∑n=p+1∞bnznanalytic in the unit discU={z:|z|<1}and satisfying the conditionzg′(z)g(z)<p+[pB+(A−B)(p−α)]z1+Bz,   z∈U,   −1≦B<A≦1,   0≦α<p.LetC(A,B,p,β,α)denote the class of functionsf(z)=zp+∑n=p+1∞anznanalytic inU, and satisfying the conditionRe{zf′(z)g(z)}>β,  z∈U,   g∈S(A,B,p,α).In this paper we determine the coefficient estimates and distortion theorems for the classC(A,B,p,β,α).