Schwarz-Pick inequalities for convex domains

Abstract

Let Ω and Π be two simply connected domains in the complex plane C, which are not equal to the whole plane C, and let A(Ω, Π) denote the set of functions f : Ω → Π analytic in Ω. Define the quantities C_n (Ω, Π) by

$C_{n}(\Omega,\Pi):=\sup\limits_{f\in A(\Omega,\Pi)}\sup\limits_{z\in \Omega} \frac{|f^{(n)}(z)|\lambda_{\Pi}(f(z))}{n!(\lambda_{\Omega}(z))^{n}},\;\; n\in \mathbb{N}$

where λ_Ω and λ_Π are the densities of the Poincaré metric in Ω and Π, respectively. We derive sharp upper bounds for |f⁽ⁿ⁾(z)| (z $\in$ Ω) and C_n(Ω, Π) if 2 ≤ n ≤ 8 and Ω is a convex domain. The detailed equality condition of the estimate on |f⁽ⁿ⁾(z)| is also given.