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

Abstract

Let $\Omega$ and $\Pi$ be two simply connected proper subdomains of the complex plane $\mathbb{C}$. We are concerned with the set $A(\Omega,\Pi)$ of functions $f: \Omega\longrightarrow\Pi$ holomorphic on $\Omega$ and we prove estimates for $|f^{(n)}(z)|, f \in A(\Omega,\Pi), z \in \Omega$, of the following type. Let $\lambda_{\Omega}(z)$ and $\lambda_{\Pi}(w)$ denote the density of the Poincaré metric with curvature $K=-4$ of $\Omega$ at $z$ and of $\Pi$ at $w$, respectively. Then for any pair $(\Omega,\Pi)$ of convex domains, $f \in A(\Omega,\Pi), z \in \Omega$, and $n\geq 2$ the inequality $$ \frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))} $$ is valid. The constant $2^{n-1}$ is best possible for any pair $(\Omega,\Pi)$ of convex domains. For any pair $(\Omega,\Pi)$, where $\Omega$ is convex and $\Pi$ linearly accessible, $f,z,n$ as above, we prove $$ \frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}. $$ The constant $2^{n-2}$ is best possible for certain admissible pairs $(\Omega,\Pi)$. These considerations lead to a new, nonanalytic, characterization of bijective convex functions $h:\Delta\to\Omega$ not using the second derivative of $h$.