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

# 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<A1, 1B<1 and 0α<p, denote by P(A,B,p,α) the class of functions p(z)=p+n=1bnzn analytic in such that P(z) ϵ P(A,B,p,α) if and only if P(z)=p+[pB+(AB)(pα)]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+1anzn 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).