We present some characterizations of Euclidean convex regions and also spherical convex regions in the Riemann sphere P in terms of the relative hyperbolic densities of the hyperbolic metrics of the given regions to C or P equipped with the Euclidean metric of the complex plane С or the spherical metric of P.  For a subregion Ω of a simply connected region X in C, we obtained analogous characterizations for hyperbolically convexity in terms of the relative hyperbolic density of the hyperbolic metric of the regions Ω to X. The notion of hyperbolic concavity relative to X for real-valued functions on Ω is also given to describe some sufficient conditions for hyperbolic convexity.