International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 47-54

Coefficient estimates for some classes of p-valent functions

M. K. Aouf

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received 6 January 1985; Revised 7 May 1985

Copyright © 1988 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.


Let Ap, where p is a positive integer, denote the class of functions f(z)=zp+n=p+1anzn which are analytic in U={z:|z|<1}.

For 0<λ1, |α|<π2, 0β<p, let Fλ(α,β,p) denote the class of functions f(z)Ap which satisfy the condition|H(f(z))1H(f(z))+1|<λforzU,where H(f(z))=eiαzf(z)f(z)βcosαipsinα(pβ)cosα.

Also let Cλ(b,p), where p is a positive integer, 0<λ<1, and b0 is any complex number, denote the class of functions g(z)Ap which satisfy the condition|H(g(z))1H(g(z))+1|<λforzU,whereH(g(z))=1+1pb(1+zg(z)g(z)p).

In this paper we obtain sharp coefficient estimates for the above mentioned classes.