Estimates for integral means of hyperbolically convex functions

We prove the Mejia–Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like $O(\log^{-2}(n)/n)$ as $(n\to\infty)$ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.