The hyperbolic metirc and bloch constants for spherically convex regions

Abstract

Let Ω denote a simply connected region on the Riemann sphere P such that Ω is convex relative to spherical geometry and Ω ≠ P. The quantity (1 +|z|²λΩ(z) is called the spherical density of the hyperbolic metric λΩ(z)|dz|on Ω. Two sharp lower bounds for the spherical density are obtained. First (1 +|z|²)λΩ(z)⩾ cosec with equality if and only if Ω is a hemisphere, where denotes the spherical distance from z to . Second, if for all ζ∈Ω then λΩ(z)⩾π/4Ø These bounds lead to covering theorems for the class K _sα of all univalent functions f in the unit disk D such that and f(D) is spherically convex. The Koebe set f ∈ K _s(α)) is the spherical disk about the origin with spherical radius (1 2) aresin α while the spherical Bloch constant for the family K _s(α)is απ4. For both covering results all extremal functions are identified.

AMS (MOS):

• 30C25
• 30C80
• 30C99