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Generalized Schwarz-Pick estimates. (English) Zbl 1012.30015
The authors obtain, among others, the following higher derivative generalization of the Schwarz-Pick inequality for analytic self maps $\varphi$ on the unit disc $D$ of the complex plane. For $n \geq 1$, $$ \text{ sup}_{z \in D} \frac{|\varphi^{(n)}(z)|(1-|z|^2)^n} {1-|\varphi(z)|^2} < \infty. $$ The proof is obtained by induction using the Faà di Bruno’s formula, as a consequence of recent characterizations of boundedness and compactness of weighted composition operators between Bloch-type spaces due to Ohno, Stroethoff and Zhao. Related work on weighted composition operators on weighted Bloch spaces is due to [{\it M. Contreras} and {\it A. G. Hernández-Díaz,} J. Aust. Math. Soc., Ser. A 69, 41-60 (2000; Zbl 0990.47018)].

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
47B33Composition operators
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