Yamashita, Shinji The Pick version of the Schwarz lemma and comparison of the Poincaré densities. (English) Zbl 0823.30014 Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, No. 2, 291-322 (1994). As the title indicates, there are two main aspects of the paper.The classical Schwarz-Pick Lemma asserts that if \(f\) is a holomorphic self-mapping of the unit disk \(\mathbb{D}\) into itself, then \[ \Gamma(z, f)= (1- | z|^ 2) | f'(z)|/(1- | f'(z)|^ 2)\leq 1 \] with strict inequality unless \(f\) is a conformal automorphism of \(\mathbb{D}\). The author refines this result by showing \(\Gamma(z, f)\leq \tanh[2d(0, z)+\tanh^{- 1} \Gamma(0, f)]\) and carefully determines all functions and points for which equality holds. Here, \(d(0, z)\) denotes the hyperbolic distance between 0 and \(z\) relative to \(\mathbb{D}\).The principle of hyperbolic metric (a consequence of the Schwarz-Pick lemma) asserts that if \(G\), \(H\) are hyperbolic regions in the complex plane \(\mathbb{C}\) with \(G\subset H\), then \(\lambda_ H(z)\leq \lambda_ G(z)\) for all \(z\in G\) with strict inequality unless \(G= H\). Here \(\lambda_ G\) \((\lambda_ H)\) denotes the density of the hyperbolic metric on \(G\) \((H)\). The author refines the principle of hyperbolic metric in several ways.One is that when \(G\) is a proper subregion of \(H\), then \(\lambda_ H(z)< \lambda_ G(z)[1- \exp(- 4d_{G, H}(z))]^{1/2}\), where \(d_{G, H}(z)\) is the hyperbolic distance relative to \(H\) from \(z\) to \(\partial G\cap H\). In this inequality the constant \(-4\) is best possible; it cannot be replaced by any constant \(C\) with \(- 4< C< 0\). Reviewer: D.Minda (Cincinnati) Cited in 8 Documents MSC: 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) Keywords:Schwarz’ lemma; hyperbolic metric PDFBibTeX XMLCite \textit{S. Yamashita}, Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, No. 2, 291--322 (1994; Zbl 0823.30014) Full Text: EuDML