Hyperbolic convexity of holomorphic level sets

We prove that the sublevel set \{z\in\mathbb{D}: k_{\mathbb{D}}(z,z_{0})-k_{\mathbb{D}}(f(z),w_{0})<\mu\} , \mu\in\mathbb{R} , is geodesically convex with respect to the Poincaré distance k_{\mathbb{D}} in the unit disk \mathbb{D} for every z_{0},w_{0}\in\mathbb{D} and every holomorphic f\colon \mathbb{D}\to\mathbb{D} if and only if \mu \le 0 . An analogous result is established also for the set \{z\in\mathbb{D}: 1-|f(z)|^{2}<\lambda(1-|z|^{2})\} , \lambda>0 . This extends a result of Solynin (2007) and solves a problem posed by Arango, Mejía and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.