On certain classes of p-Valent functions

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.