Punishing factors and Chua's conjecture

Abstract

Let $\Omega $ and $\Pi $ be two simply connected domains in the complex plane $ \mathds{C}$ which are not equal to the whole plane $\mathds{C}$. We are concerned with the set $A(\Omega,\Pi)$ of functions $f: \Omega\to\Pi$ 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 Poincar\'{e} metric of $\Omega$ at $z$ and of $\Pi$ at $w$, respectively. Then for any pair $(\Omega,\Pi)$ where $\Omega$ is convex, $f\in A(\Omega,\Pi), z \in \Omega$, and $n\geq 2$ the inequality \[ \frac{|f^{(n)}(z)|}{n!}\leq (n+1) 2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))} \] is valid.

For functions $f\in A(\Omega,\Pi)$, which are injective on $\Omega$, the validity of above inequality was conjectured by Chua in 1996.