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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 $ℂ$ which are not equal to $ℂ$. We denote by ${\lambda }_{{\Omega }}$ and ${\lambda }_{{\Pi }}$ the densities of the hyperbolic metrics in ${\Omega }$ and ${\Pi }$ respectively. Let $A\left({\Omega },{\Pi }\right)$ denote the set of functions $f:{\Omega }\to {\Pi }$ analytic in ${\Omega }$. The authors consider the quantities

${M}_{n}\left(z,{\Omega },{\Pi }\right):=sup\left\{\frac{|{f}^{\left(n\right)}\left(z\right)|}{n!}\frac{{\lambda }_{{\Pi }}\left(f\left(z\right)\right)}{{\left({\lambda }_{{\Omega }}\left(z\right)\right)}^{n}}:f\in A\left({\Omega },{\Pi }\right)\right\},n\in ℕ,z\in {\Omega },$
${C}_{n}\left({\Omega },{\Pi }\right):=sup\left\{{M}_{n}\left(z,{\Omega },{\Pi }\right):z\in {\Omega }\right\}·$

It follows by the Schwarz-Pick lemma that ${C}_{1}\left({\Omega },{\Pi }\right)=1$ for any pair of simply connected domains. The quantity ${C}_{n}\left({\Omega },{\Pi }\right)$ 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 ${\Pi }$,

${M}_{n}\left(z,{\Delta },{\Pi }\right)={\left(1+|z|\right)}^{n-1}·$

It follows that ${C}_{n}\left({\Delta },{\Pi }\right)={2}^{n-1}$ for convex domain ${\Pi }$. Furthermore, they show that ${C}_{n}\left({\Omega },{\Pi }\right)\le {4}^{n-1}$ holds for arbitrary simply connected domains whereas the inequality ${2}^{n-1}\le {C}_{n}\left({\Omega },{\Pi }\right)$ is proved only under some technical restriction upon ${\Omega }$ and ${\Pi }$.

MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C55 General theory of univalent and multivalent functions