A simply connected region
Ω in the complex plane ℂ with smooth boundary ∂Ω is called k-convex
(k > 0) if k(z,∂Ω) ≥ k for all z ∈ Ω, where k(z,∂Ω) denotes the euclidean
curvature of ∂Ω at the point z. A different definition is used when ∂Ω is
not smooth. We present a study of the hyperbolic geometry of k-convex
regions. In particular, we obtain sharp lower bounds for the density λΩ of the
hyperbolic metric and sharp information about the euclidean curvature and
center of curvature for a hyperbolic geodesic in a k-convex region. We give
applications of these geometric results to the family K(k,α) of all conformal
mappings f of the unit disk 𝔻 onto a k-convex region and normalized by
f(0) = 0 and f′(0) = α > 0. These include precise distortion and covering
theorems (the Bloch-Landau constant and the Koebe set) for the family
K(k,α).
