Hyperbolic convexity of holomorphic level sets

We prove that the sublevel set $\{z\in \mathbb{D}\text{\hspace{0.17em}⁣}:k_{\mathbb{D}}(z,z_0)-k_{\mathbb{D}}(f(z),w_0)<\mu \}$, $\mu \in \mathbb{R}$, 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 $f:\mathbb{D}\to \mathbb{D}$ if and only if $\mu ⩽0$. An analogous result is established also for the set $\{z\in \mathbb{D}\text{\hspace{0.17em}⁣}:1-\mathrm{\mid }f(z){\mathrm{\mid }}^2<\lambda (1-\mathrm{\mid }z{\mathrm{\mid }}^2)\}$, $\lambda >0$. 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.