Some inequalities for the Poincaré metric of plane domains
The paper gives an alternative proof of the following important recent result of \textit{F. P. Gardiner} and \textit{N. Lakic} [Ann. Math. (2) 154, No. 2, 245--267 (2001; Zbl 1045.30026)]. Let \(\Omega\) be a connected open set in \({\mathbb C}\) whose boundary contains at least two points, and let \(p:{\mathbb D} \to \Omega\) be the holomorphic universal cover of \(\Omega\) by the unit disc \({\mathbb D}\). The Poincaré metric \(\rho(z)| dz| \) is defined by \(\rho(p(z)) | p'(z)| = 1/(1-| z| ^2)\), for each \(z \in {\mathbb D}\). Furthermore, let \(\lambda_{a,b} (w) | dw| \) be the hyperbolic metric of the twice punctured plane \({\mathbb C} \setminus \{a,b\}\), and define \(\sigma(z) := \sup_{a,b \in \partial \Omega} \lambda_{a,b} (z)\). We then have that there exists a universal constant \(C>0\) not depending on \(\Omega\) such that \[ \sigma (z) \leq \rho(z) \leq C \sigma(z), \quad\text{for each } z \in \Omega. \] The paper gives an explicit estimate for the constant \(C\). Namely, \(C\) can be taken to be \[ C= \left(\pi^3 + (\Gamma(1/4))^4\right) / (2 \pi^2) \approx 10.3246. \] Also, the paper gives further interesting concrete estimates for the hyperbolic distance near a boundary point. These are then used to obtain refinements of \textit{J. E. Littlewood}'s Theorem [Proceedings L. M. S. (2) 23; V-IX, 481-519. (1925; JFM 51.0247.03)].
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