On quasi-convex functions and related topics

Abstract

LetSbe the class of functionsfwhich are analytic and univalent in the unit discEwithf(0)=0,f′(0)=1. LetC,S*andKbe the classes of convex, starlike and close-to-convex functions respectively. The classC*of quasi-convex functions is defined as follows:Letfbe analytic inEandf(0),f′(0)=1. Thenf ϵ C*if and only if there exists ag ϵ Csuch that, forz ϵ ERe(zf′(z))′g′(z)>0.In this paper, an up-to-date complete study of the classC*is given. Its basic properties, its relationship with other subclasses ofS, coefficient problems, arc length problem and many other results are included in this study. Some related classes are also defined and studied in some detail.