Strongly hyperbolically convex functions

Let C(w1,w2,w3) denote the circle in C through w1,w2,w3 and let w1w2 denote one of the two arcs between w1,w2 belonging to C(w1,w2,w3). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w1,w2,w3 ∈ Ω, with w1 = w2, the arc w1w2 of the circle C(w1,w2,−1/w3 ) which does not contain −1/w3 lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w1,w2,w3 ∈ G, with w1 = w2, the arc w1w2 in D of the circle C(w1,w2, 1/w3 ) is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained. © 2007 Elsevier Inc. All rights reserved