Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T19:02:58.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius

Published online by Cambridge University Press:  20 November 2018

Dimitrios Betsakos  and
Stamatis Pouliasis
Dimitrios Betsakos
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail: [email protected]@math.auth.gr
Stamatis Pouliasis
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail: [email protected]@math.auth.gr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove variants of Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we examine when a similar monotonicity property holds for the hyperbolic metric.

Keywords

30C8030F4531A15condenser capacityinner radiushyperbolic metricSchwarz's lemma
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 241 - 250
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Beardon, A. F. and Minda, D., The hyperbolic metric and geometric function theory. In: Proceedings of the InternationalWorkshop on Quasiconformal Mappings and their Applications, Narosa Publishing House, New Delhi, 2007, 9–56. Google Scholar
[2] Beckenbach, E. F., A relative of the lemma of Schwarz. Bull. Amer. Math. Soc. 44(1938), 698–707. http://dx.doi.org/10.1090/S0002-9904-1938-06845-0 Google Scholar
[3] Betsakos, D., Geometric versions of Schwarz's lemma for quasiregular mappings. Proc. Amer. Math. Soc. 139(2011), 1397–1407. http://dx.doi.org/10.1090/S0002-9939-2010-10604-4 Google Scholar
[4] Betsakos, D., Multi-point variations of Schwarz lemma with diameter and width conditions. Proc. Amer. Math. Soc. 139(2011), no. 11, 4041–4052. http://dx.doi.org/10.1090/S0002-9939-2011-10954-7 Google Scholar
[5] Burckel, R. B., Marshall, D. E., Minda, D., P. Poggi-Corradini and Ransford, T. J., Area, capacity and diameter versions of Schwarz's lemma. Conform. Geom. Dyn. 12(2008), 133–152. http://dx.doi.org/10.1090/S1088-4173-08-00181-1 Google Scholar
[6] Carroll, T. and Ratzkin, J., Isoperimetric inequalities and variations on Schwarz's lemma. Preprint, 2010.Google Scholar
[7] Conway, J. B., Functions of One Complex Variable. Graduate Texts in Mathematics 11, Springer-Verlag, New York–Heidelberg, 1973.Google Scholar
[8] Dubinin, V. N., Symmetrization in the geometric theory of functions of a complex variable. (Russian) Uspekhi Mat. Nauk 49(1994), 3–76. translation in Russian Math. Surveys 49(1994), 1–79. Google Scholar
[9] Dubinin, V. N., Geometric versions of the Schwarz lemma and symmetrization. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 383(2010), Analiticheskaya Teoriya Chisel i Teoriya Funktsii. 25, 63–76. 205–206. Google Scholar
[10] Hayman, W. K., Multivalent Functions. Second edition, Cambridge Tracts in Mathematics 110, Cambridge University Press, Cambridge, 1994.Google Scholar
[11] Hayman, W. K., Subharmonic Functions, Vol. 2. London Mathematical Society Monographs 20, Academic Press, London, 1989.Google Scholar
[12] Julia, G., Sur les moyennes des modules de fonctions analytiques. Bull. Sci. Math. 51(1927), 198–214. Google Scholar
[13] Laugesen, R. and Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mappings. J. Funct. Anal. 155(1998), 64–108. http://dx.doi.org/10.1006/jfan.1997.3222 Google Scholar
[14] Pólya, G. and Szegö, G., Problems and Theorems in Analysis. I. Springer, 1978.Google Scholar
[15] Pouliasis, S., Condenser capacity and meromorphic functions. Comput. Methods Funct. Theory 11(2011), 237–245. Google Scholar
[16] Xiao, J. and Zhu, K., Volume integral means of holomorphic functions. Proc. Amer. Math. Soc. 139(2011), 1455–1465. http://dx.doi.org/10.1090/S0002-9939-2010-10797-9 Google Scholar