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Schwarz-Pick inequalities for derivatives of arbitrary order. (English) Zbl 1018.30018

Let \(\Omega\) and \(\Pi\) be two simply connected domains in the complex plane \(\mathbb C\) which are not equal to \(\mathbb C\). We denote by \(\lambda_\Omega\) and \(\lambda_\Pi\) the densities of the hyperbolic metrics in \(\Omega\) and \(\Pi\) respectively. Let \(A(\Omega,\Pi)\) denote the set of functions \(f:\Omega\to \Pi\) analytic in \(\Omega\). The authors consider the quantities \[ M_n(z,\Omega,\Pi):=\sup \left\{\frac{|f^{(n)}(z)|}{n!}\frac{\lambda_\Pi(f(z))}{(\lambda_\Omega(z))^n}:f\in A(\Omega,\Pi)\right\}, n\in \mathbb N, z\in\Omega, \]
\[ C_n(\Omega,\Pi):=\sup\{M_n(z,\Omega,\Pi):z\in\Omega\}. \] It follows by the Schwarz-Pick lemma that \(C_1(\Omega,\Pi)=1\) for any pair of simply connected domains. The quantity \(C_n(\Omega,\Pi)\) has been computed in some special cases by St. Ruscheweyh (for example, when \(\Omega\) is the unit disk \(\Delta\) and \(\Pi\) is a half-plane).
The authors of the article under review show that for any convex domain \(\Pi\), \[ M_n(z,\Delta,\Pi)=(1+|z|)^{n-1}. \] It follows that \(C_n(\Delta,\Pi)=2^{n-1}\) for convex domain \(\Pi\). Furthermore, they show that \(C_n(\Omega,\Pi)\leq 4^{n-1}\) holds for arbitrary simply connected domains whereas the inequality \(2^{n-1}\leq C_n(\Omega,\Pi)\) is proved only under some technical restriction upon \(\Omega\) and \(\Pi\).

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C55 General theory of univalent and multivalent functions of one complex variable
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