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In a series of papers ([MM],[MM 9],[M Mg]), Mejia and Minda studied hyperbolic geometry in regions which are k-convex relative to euclidean, spherical or hyperbolic geometry. In addition, they investigated conformal maps of the unit disk D=(z: lzl< 1) onto such regions. The cases of euclidean and spherical h-convexity were very similar; corresponding results had analogous geometric proofs. The case of hyperbolic h-convexity presented several intriguing differences. First, there were often differences between the cases 0< k< 2 and 2 sk, while no such differences occured in either the euclidean or spherical situation. Second, some of the geometric methods that worked in the first two papers did not generalize to the hyperbolic case, so the third paper did not contain analogs of all of the results of the first two papers. Recently, we [MMM] found a new geometric approach to euclidean k-convex functions. This ethod …