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 [7], where they initiated the study of an analytic

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... fixed point function of ϕ and its applications to Probability Theory; see [7], [8]. For the case when the function g(z) = z/ϕ(z) is univalent in D, this theorem was proved in [7]. The fixed point function of ϕ is defined in [7] as the solution to the functional equation wϕ(f (w)) = f (w) for w ∈ D.