A note on Schwarz's lemma (Q2040232)

The author uses the classical ideas related to the Alhlfors-Schwarz lemma and involving conformal metrics \(\rho\) and their Gaussian curvature \(k_\rho\) to prove the following result: Suppose that \(\rho_1,\rho_2\) are two metrics with strictly negative curvature on a planar domain \(\Omega\). Assume that \(\rho_2\) is strictly positive and continuous on \(\Omega\), and that \(\rho_1/\rho_2\) attains its maximum inside \(\Omega\). Then for every \(z\in\Omega\), \[ \frac{\rho_1(z)}{\rho_2(z)}\leq \left \|\frac{k_{\rho_1}(z)}{k_{\rho_2}(z)}\right \|_\infty^{1/2}. \] The author gives an application of this result for extremal metrics on star-shaped domains.