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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 $\Bbb C$ which are not equal to $\Bbb C$. 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 $$ M_n(z,\Omega,\Pi):=\sup \left\{\frac{|f^{(n)}(z)|}{n!}\frac{\lambda_\Pi(f(z))}{(\lambda_\Omega(z))^n}:f\in A(\Omega,\Pi)\right\}, n\in \Bbb N, z\in\Omega,$$ $$ C_n(\Omega,\Pi):=\sup\{M_n(z,\Omega,\Pi):z\in\Omega\}. $$ 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 $\Pi$.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C55General theory of univalent and multivalent functions
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