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The Pick version of the Schwarz lemma and comparison of the Poincaré densities. (English) Zbl 0823.30014

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 \(- 4< C< 0\).

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)
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