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We answer a question raised by D. Mejía and Ch. Pommerenke by showing that the analytic fixed point function is hyperbolically convex in the unit disc. Let D be the unit disc in C and let ϕ : D → D be an analytic map. The goal of this note is to prove the following. It is hyperbolically strictly convex unless ϕ is a Möbius map, in which case F is a hyperbolic half-plane. Thus this theorem solves Problem 1 posed by D. Mejía and Ch. Pommerenke in , where they initiated the study of an analyticdoi:10.1090/s0002-9939-06-08661-8 fatcat:kvcmf2uqqzf4tk4z5w5p2iqhbu