Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Hyperbolic geometric versions of Schwarz's lemma


Author: Dimitrios Betsakos
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
Published electronically: November 1, 2013
MathSciNet review: 3126908
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD). A Wiley-Interscience Publication. MR 1462077 (98h:30033)
  • [2] Rauno Aulaskari and Huaihui Chen, Area inequality and $ Q_p$ norm, J. Funct. Anal. 221 (2005), no. 1, 1-24. MR 2124895 (2005k:30066), https://doi.org/10.1016/j.jfa.2004.12.007
  • [3] A. F. Beardon and D. Minda, A multi-point Schwarz-Pick lemma, J. Anal. Math. 92 (2004), 81-104. MR 2072742 (2005f:30044), https://doi.org/10.1007/BF02787757
  • [4] A. F. Beardon and D. Minda, The hyperbolic metric and geometric function theory, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9-56. MR 2492498 (2011c:30108)
  • [5] Dimitrios Betsakos, Geometric versions of Schwarz's lemma for quasiregular mappings, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1397-1407. MR 2748432 (2011k:30030), https://doi.org/10.1090/S0002-9939-2010-10604-4
  • [6] Dimitrios Betsakos, Multi-point variations of the Schwarz lemma with diameter and width conditions, Proc. Amer. Math. Soc. 139 (2011), no. 11, 4041-4052. MR 2823049, https://doi.org/10.1090/S0002-9939-2011-10954-7
  • [7] Dimitrios Betsakos and Stamatis Pouliasis, Versions of Schwarz's lemma for condenser capacity and inner radius, Canad. Math. Bull. 56 (2013), no. 2, 241-250. MR 3043051, https://doi.org/10.4153/CMB-2011-189-8
  • [8] Robert B. Burckel, Donald E. Marshall, David Minda, Pietro Poggi-Corradini, and Thomas J. Ransford, Area, capacity and diameter versions of Schwarz's lemma, Conform. Geom. Dyn. 12 (2008), 133-152. MR 2434356 (2010j:30050), https://doi.org/10.1090/S1088-4173-08-00181-1
  • [9] G. Cleanthous, Monotonicity theorems for analytic functions centered at infinity. Proc. Amer. Math. Soc. (to appear).
  • [10] V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3-76 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054), https://doi.org/10.1070/RM1994v049n01ABEH002002
  • [11] V. N. Dubinin, Geometric versions of the Schwarz lemma and symmetrization, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 383 (2010), no. Analiticheskaya Teoriya Chisel i Teoriya Funktsii. 25, 63-76, 205-206 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 178 (2011), no. 2, 150-157. MR 2749342 (2011k:30031), https://doi.org/10.1007/s10958-011-0542-0
  • [12] V. N. Dubinin, On the preservation of conformal capacity under a mapping by meromorphic functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 392 (2011), no. Analiticheskaya Teoriya Chisel i Teoriya Funktsii. 26, 67-73, 219 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 184 (2012), no. 6, 699-702. MR 2870219 (2012k:31002), https://doi.org/10.1007/s10958-012-0891-3
  • [13] Alexander Fryntov and John Rossi, Hyperbolic symmetrization and an inequality of Dynkin, Entire functions in modern analysis (Tel-Aviv, 1997) Israel Math. Conf. Proc., vol. 15, Bar-Ilan Univ., Ramat Gan, 2001, pp. 103-115. MR 1890533 (2003b:30041)
  • [14] F. W. Gehring, Inequalities for condensers, hyperbolic capacity, and extremal lengths, Michigan Math. J. 18 (1971), 1-20. MR 0285697 (44 #2915)
  • [15] W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. MR 1310776 (96f:30003)
  • [16] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486 (13,270d)
  • [17] Ch. Pommerenke, On hyperbolic capacity and hyperbolic length, Michigan Math. J. 10 (1963), 53-63. MR 0148882 (26 #6379)
  • [18] Stamatis Pouliasis, Condenser capacity and meromorphic functions, Comput. Methods Funct. Theory 11 (2011), no. 1, 237-245. MR 2816955, https://doi.org/10.1007/BF03321800
  • [19] M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894 (22 #5712)
  • [20] Jie Xiao, Isoperimetry for semilinear torsion problems in Riemannian two-manifolds, Adv. Math. 229 (2012), no. 4, 2379-2404. MR 2880225, https://doi.org/10.1016/j.aim.2012.01.009
  • [21] Jie Xiao and Kehe Zhu, Volume integral means of holomorphic functions, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1455-1465. MR 2748439 (2012b:32012), https://doi.org/10.1090/S0002-9939-2010-10797-9
  • [22] Shinji Yamashita, Length and area inequalities for the derivative of a bounded and holomorphic function, Bull. Austral. Math. Soc. 30 (1984), no. 3, 457-462. MR 766803 (86a:30040), https://doi.org/10.1017/S0004972700002173

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 30C80, 30C85, 30F45, 30H05

Retrieve articles in all journals with MSC (2010): 30C80, 30C85, 30F45, 30H05


Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

DOI: https://doi.org/10.1090/S1088-4173-2013-00260-9
Keywords: Holomorphic function, Schwarz lemma, hyperbolic metric, hyperbolic area, hyperbolic capacity, hyperbolic diameter, condenser, symmetrization.
Received by editor(s): June 20, 2013
Received by editor(s) in revised form: September 14, 2013
Published electronically: November 1, 2013
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society