Spherical convexity and hyperbolic metric

Let $Omega$ be a domain in $mathbb{C}$ with hyperbolic metric $lambda_Omega(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $Omega$ is (Euclidean) convex if and only if $d(z,partialOmega)lambda_Omega(z)ge1/2$ for $zinOmega,$ where $d(z,partialOmega)$ denotes the Euclidean distance from $z$ to the boundary $partialOmega.$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.