International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 4, Pages 733-744
doi:10.1155/S0161171287000838

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

M. K. Aouf1,2

1Dept. of Maths., Faculty of Science, University of Mansoura, Mansoura, Egypt
2Dept. of Mathematics, Faculty of Science, University of Qatar, P.O. Box 2713, Doha, Qatar

Received 9 October 1986

Copyright © 1987 M. K. Aouf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).