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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, there is a positive constant ${M}_{n,r}$ such that for each analytic self-map $\varphi$ of the unit disc $D$,

$\frac{{\left(1-|z|}^{2}{\right)}^{n}|{\varphi }^{\left(n\right)}\left(z\right)|}{1-{|\varphi \left(z\right)|}^{2}}\le {M}_{n,r}\underset{\rho \left(w,z\right)

for $z$ in $D$, where $\rho \left(w,z\right)=|\frac{w-z}{1-\overline{w}z}|$ 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, etc. (one complex variable)