Strongly hyperbolically convex functions

Let C(w1,w2,w3) denote the circle in through w1,w2,w3 and let 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 of the circle which does not contain 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 in D of the circle is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.