Stretching convex domains and the hyperbolic metric

It is shown that the log-convexity of the density of the hyperbolic metric in a convex planar domain leads to a pointwise comparison between the density of the hyperbolic metric in a convex domain D and that in a domain obtained by stretching D. Applications of this result are given, including estimates for the density of the hyperbolic metric in the domain interior to an ellipse and a lower bound for the density of the hyperbolic metric in a convex domain in terms of the density in a comparison strip. Connections are made with the convexity of related functions on convex regions in space.