On Ozaki's condition for p-valency

Abstract

Let f be an analytic function in a convex domain D⊂C. A well-known theorem of Ozaki states that if f is analytic in D, and is given by f(z)=zp+∑n=p+1∞anzn for z∈D, andRe{eiαf(p)(z)}>0,(z∈D), for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk D={z∈C:|z|<1} to be p-valent in D, and to give an improvement to Ozaki's sufficient condition for p-valence when z∈D.