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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 $D$, $$ \frac{(1-\vert z\vert ^2)^n\vert \varphi^{(n)}(z)\vert }{1-\vert \varphi(z)\vert ^2} \le M_{n,r} \max_{\rho(w,z)<r} \frac{(1-\vert w\vert ^2)\vert \varphi^{'}(w)\vert }{1-\vert \varphi(w)\vert ^2} $$ for $z$ in $D$, where $\rho(w,z)=\big\vert \frac{w-z}{1-\bar wz} \big\vert $ 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.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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