International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 55-64

On certain classes of p-Valent functions

M. K. Aouf

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

Received 20 February 1985

Copyright © 1986 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 Vkλ(α,b,p) (k2, b0 is any complex number, 0α<p and |λ|<π/2) denote the class of functions f(z)=zp+n=p+1anzn analytic in U={z:|z|<1} having (p1) critical points in U and satisfyinglimr1sup02π|Re{eiλ[p+1b(1+zf(z)f(z)p)]αcosλ}pα|dθkπcosλ.In this paper we generalize both those functions f(z) which are p-valent convex of order α, 0α<p, with bounded boundary rotation and those p-valent functions f(z) for which zf(z)/p is λ-spirallike of order α, 0α<p.