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Let Ω and Π be two simply connected proper subdomains of the complex plane C. We are concerned with the set A(Ω,Π) of functions f : Ω −→ Π holomorphic on Ω and we prove estimates for |f (n)(z)|, f ∈ A(Ω,Π), z ∈ Ω, of the following type. Let λΩ(z) and λΠ(w) denote the density of the Poincaré metric with curvature K = −4 of Ω at z and of Π at w, respectively. Then for any pair (Ω,Π) of convex domains, f ∈ A(Ω,Π), z ∈ Ω, and n ≥ 2 the inequality |f (n)(z)| n! ≤ 2n−1 (λΩ(z)) n λΠ(f(z)) is valid. The constant 2n−1 is best possible for any pair (Ω,Π) of convex domains. For any pair (Ω,Π), where Ω is convex and Π linearly accessible, f, z, n as above, we prove |f (n)(z)| (n + 1)! ≤ 2n−2 (λΩ(z)) n λΠ(f(z)) . The constant 2n−2 is best possible for certain admissible pairs (Ω,Π). These considerations lead to a new, nonanalytic, characterization of bijective convex functions h : ∆ → Ω not using the second derivative of h. Let Ω and Π be two simply connected proper subdomains of the complex plane C and A(Ω, Π) = {f : Ω → Π | f holomorphic}. Furthermore, let λΩ(z), z ∈ Ω, and λΠ(w), w ∈ Π, denote the density of the Poincaré metric with curvature K = −4 at z ∈ Ω and w ∈ Π, respectively. 2000 Mathematics Subject Classification: 30C50, 30C45, 30D50.

The punishing factors for convex pairs are $2^{n-1}$