Ghatage, Pratibha; Zheng, Dechao Hyperbolic derivatives and generalized Schwarz-Pick estimates. (English) Zbl 1050.30016 Proc. Am. Math. Soc. 132, No. 11, 3309-3318 (2004). 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. Reviewer: Lou Zengjian (Shantou Guangdong) Cited in 18 Documents MSC: 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination Keywords:generalized Schwarz-Pick estimate; hyperbolic derivative; pseudo-hyperbolic distance PDFBibTeX XMLCite \textit{P. Ghatage} and \textit{D. Zheng}, Proc. Am. Math. Soc. 132, No. 11, 3309--3318 (2004; Zbl 1050.30016) Full Text: DOI