Koebe sets for the class of functions convex in two directions

Abstract

In this paper, we consider a class Kα of all functions f univalent in the unit disk ∆ that are normalized by f (0) = f ′ (0) − 1 = 0 while the sets f (∆) are convex in two symmetric directions: e iαπ/2 and e −iαπ/2 , α ∈ [0, 1] . It means that the intersection of f (∆) with each straight line having the direction e iαπ/2 or e −iαπ/2 is either a compact set or an empty set. We find the Koebe set for Kα . Moreover, we perform the same operation for functions in K β,γ , i.e. for functions that are convex in two fixed directions: e iβπ/2 and e iγπ/2 , −1 ≤ β ≤ γ ≤ 1 .