## Hyperbolic geometric versions of Schwarzâ€™s lemma

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- by Dimitrios Betsakos PDF
- Conform. Geom. Dyn.
**17**(2013), 119-132 Request permission

## Abstract:

Let $f$ be a holomorphic self-map of the unit disk $\mathbb {D}$. We prove monotonicity theorems which involve the hyperbolic area, the hyperbolic capacity, and the hyperbolic diameter of the images under $f$ of hyperbolic disks in $\mathbb {D}$. These theorems lead to distortion and modulus growth theorems that generalize the classical lemma of Schwarz and to geometric estimates for the density of the hyperbolic metric.## References

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## Additional Information

**Dimitrios Betsakos**- Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
- MR Author ID: 618946
- Email: betsakos@math.auth.gr
- Received by editor(s): June 20, 2013
- Received by editor(s) in revised form: September 14, 2013
- Published electronically: November 1, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**17**(2013), 119-132 - MSC (2010): Primary 30C80, 30C85, 30F45, 30H05
- DOI: https://doi.org/10.1090/S1088-4173-2013-00260-9
- MathSciNet review: 3126908