# Analytic functions & convexity

It is known (Scwhartz-Pick) that holomorphic functions are distance non-increasing in the hyperbolic metric.Does it imply that they preserve convexity in the hyperbolic metric? How about the converse?

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No. Take a convex set and use an exponential function $e^{2\pi i z/c}$, where $c$ is the difference between two points in the set. The image of the set is no longer simply connected, and thus cannot be convex in a Riemannian manifold with no closed geodesics like the hyperbolic disc (which I assume is what you are talking about here). You may need to scale the exponential function down by a constant to fit in the disc, this is obviously immaterial