Article contents
Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius
Published online by Cambridge University Press: 20 November 2018
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.
- Type
- Research Article
- Information
- 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
[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
- 5
- Cited by