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

Published online by Cambridge University Press:  20 November 2018

Dimitrios Betsakos
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail:
Stamatis Pouliasis
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece e-mail:
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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.

Research Article
Copyright © Canadian Mathematical Society 2013


[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, 956. Google Scholar
[2] Beckenbach, E. F., A relative of the lemma of Schwarz. Bull. Amer. Math. Soc. 44(1938), 698707. Google Scholar
[3] Betsakos, D., Geometric versions of Schwarz's lemma for quasiregular mappings. Proc. Amer. Math. Soc. 139(2011), 13971407. Google Scholar
[4] Betsakos, D., Multi-point variations of Schwarz lemma with diameter and width conditions. Proc. Amer. Math. Soc. 139(2011), no. 11, 40414052. 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), 133152. 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), 376. translation in Russian Math. Surveys 49(1994), 179. 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, 6376. 205206. 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), 198214. Google Scholar
[13] Laugesen, R. and Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mappings. J. Funct. Anal. 155(1998), 64108. 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), 237245. Google Scholar
[16] Xiao, J. and Zhu, K., Volume integral means of holomorphic functions. Proc. Amer. Math. Soc. 139(2011), 14551465. Google Scholar