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Hyperbolic derivatives and generalized Schwarz-Pick estimates. (English) Zbl 1050.30016

In this paper, by using a formula of Faà di Bruno for the \(n\)th derivative of composition of two functions, the authors obtain the following generalized Schwarz-Pick estimate: for each positive integer \(n\) and each number \(0<r<1\), there is a positive constant \(M_{n,r}\) such that for each analytic self-map \(\varphi\) of the unit disc \(D\),
\[ \frac{(1-| z| ^2)^n| \varphi^{(n)}(z)| }{1-| \varphi(z)| ^2} \leq M_{n,r} \max_{\rho(w,z)<r} \frac{(1-| w| ^2)| \varphi^{'}(w)| }{1-| \varphi(w)| ^2} \]
for \(z\) in \(D\), where \(\rho(w,z)=\big| \frac{w-z}{1-\bar wz} \big| \) is the pseudo-hyperbolic distance of \(z\) and \(w\) in \(D\). By means of those estimates the authors show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric.

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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