Kodai Mathematical Journal

Schwarz-Pick inequalities for convex domains

Jian-Lin Li

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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 Cn (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 λΩ and λΠ 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 Cn(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.

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Kodai Math. J., Volume 30, Number 2 (2007), 252-262.

First available in Project Euclid: 3 July 2007

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Li, Jian-Lin. Schwarz-Pick inequalities for convex domains. Kodai Math. J. 30 (2007), no. 2, 252--262. doi:10.2996/kmj/1183475516. https://projecteuclid.org/euclid.kmj/1183475516

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