Strongly hyperbolically convex functions

[摘要] Let C (w(1), w(2), w(3)) denote the circle in C through w(1), w(2), w(3) and let w(1)w(2) denote one of the two arcs between w(1), w(2) belonging to C(w(1), w(2), w(3)). We prove that a domain ohm in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w(1), w(2), w(3) ohm, with w(1) not equal w(2), the arc w(1)w(2) of the circle C(w(1), w(2), -1 root w(3)) which does not contain -1 root w(3) lies in ohm. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w(1), w(2), w(3) is an element of G, with w(1) not equal w(2), the arc w(1)w(2) in D of the circle C(w(1), w(2), 1 root w(3)) is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-11-15

[关键词] strong hyperbolic convexity;spherical convexity;hyperbolic metric