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 $f:\Omega ⟶\Pi$ 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 $n\ge 2$ the inequality

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

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 $h$.