International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 55-64
doi:10.1155/S0161171286000078
On certain classes of p-Valent functions
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.
Abstract
Let Vkλ(α,b,p) (k≥2, b≠0 is any complex number, 0≤α<p and |λ|<π/2) denote the class of functions f(z)=zp+∑n=p+1∞anzn analytic in U={z:|z|<1} having (p−1) critical points in U and satisfyinglimr→1−sup∫02π|Re{eiλ[p+1b(1+zf″(z)f′(z)−p)]−αcosλ}p−α|dθ≤kπcosλ.In this paper we generalize both those functions f(z) which are p-valent convex of order α, 0≤α<p, with bounded boundary rotation and those p-valent functions f(z) for which zf′(z)/p is λ-spirallike of order α, 0≤α<p.