Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Ahlfors-Beurling conformal invariant and relative capacity of compact sets


Authors: Vladimir N. Dubinin and Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 142 (2014), 3865-3879
MSC (2010): Primary 30C85
DOI: https://doi.org/10.1090/S0002-9939-2014-12125-3
Published electronically: July 10, 2014
MathSciNet review: 3251726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a given domain $ D$ in the extended complex plane $ \overline {\mathbb{C}}$ with an accessible boundary point $ z_0 \in \partial D$ and for a subset $ E \subset {D},$ relatively closed w.r.t. $ D,$ we define the relative capacity $ {\rm rel cap}{\,}E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $ r(D\setminus E,z)/r(D, z)$ when $ z$ approaches the point $ z_0\,.$ Here $ r(G,z)$ denotes the inner radius at $ z$ of the connected component of the set $ G$ containing the point $ z\,.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $ D$ is the upper half plane and $ z_0=\infty $, the capacity $ {\rm rel cap}{\,}E$ coincides with the well-known half-plane capacity $ {\rm hcap}{\,}E\,.$ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.


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

  • [A] Lars Valerian Ahlfors, Collected papers. Vol. 1. 1929-1955, edited with the assistance of Rae Michael Shortt. Contemporary Mathematicians, Birkhäuser Boston, Mass., 1982. MR 688648 (84k:01066a)
  • [AB] Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101-129. MR 0036841 (12,171c)
  • [D1] 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
  • [D2] V. N. Dubinin, Lower bounds for the half-plane capacity of compact sets and symmetrization, Mat. Sb. 201 (2010), no. 11, 77-88 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 11-12, 1635-1646. MR 2768554 (2011m:30059), https://doi.org/10.1070/SM2010v201n11ABEH004125
  • [D3] V. N. Dubinin, On the boundary values of the Schwarzian derivative of a regular function, Mat. Sb. 202 (2011), no. 5, 29-44 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 5-6, 649-663. MR 2841516 (2012f:30047), https://doi.org/10.1070/SM2011v202n05ABEH004159
  • [H] W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. MR 1310776 (96f:30003)
  • [He] Joseph Hersch, On the reflection principle and some elementary ratios of conformal radii, J. Analyse Math. 44 (1984/85), 251-268. MR 801297 (87h:30020), https://doi.org/10.1007/BF02790200
  • [LLN] Steven Lalley, Gregory Lawler, and Hariharan Narayanan, Geometric interpretation of half-plane capacity, Electron. Commun. Probab. 14 (2009), 566-571. MR 2576752 (2011b:60332), https://doi.org/10.1214/ECP.v14-1517
  • [L1] Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR 2129588 (2006i:60003)
  • [L2] G. Lawler, Schramm-Loewner evolution (SLE), Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., Providence, RI, 2009, pp. 231-295. MR 2523461 (2011d:60244)
  • [L3] Gregory F. Lawler, Conformal invariance and 2D statistical physics, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 35-54. MR 2457071 (2010a:60283), https://doi.org/10.1090/S0273-0979-08-01229-9
  • [M1] Moshe Marcus, Transformations of domains in the plane and applications in the theory of functions, Pacific J. Math. 14 (1964), 613-626. MR 0165093 (29 #2382)
  • [M2] Moshe Marcus, Radial averaging of domains, estimates for Dirichlet integrals and applications, J. Analyse Math. 27 (1974), 47-78. MR 0477029 (57 #16573)
  • [O] M. Ohtsuka, Dirichlet problem, extremal length, and prime ends, Van Nostrand Reinhold, 1970.
  • [PS] 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)
  • [Pom] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
  • [R] Heinz Renggli, An inequality for logarithmic capacities, Pacific J. Math. 11 (1961), 313-314. MR 0155992 (27 #5925)
  • [RW] S. Rohde and C. Wong, Half-plane capacity and conformal radius, Proc. Amer. Math. Soc. 142 (2014), no. 3, 931-938. MR 3148527
  • [S] David Shoikhet, Another look at the Burns-Krantz theorem, J. Anal. Math. 105 (2008), 19-42. MR 2438420 (2009h:30040), https://doi.org/10.1007/s11854-008-0030-8
  • [TV] Roberto Tauraso and Fabio Vlacci, Rigidity at the boundary for holomorphic self-maps of the unit disk, Complex Variables Theory Appl. 45 (2001), no. 2, 151-165. MR 1909431 (2003e:30039)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C85

Retrieve articles in all journals with MSC (2010): 30C85


Additional Information

Vladimir N. Dubinin
Affiliation: Far Eastern Federal University, Vladivostok, Russia
Address at time of publication: Institute of Applied Mathematics, Vladivostok, Russia
Email: dubinin@iam.dvo.ru

Matti Vuorinen
Affiliation: Department of Mathematics and Statistics, University of Turku, Turku 20014, Finland
Email: vuorinen@utu.fi

DOI: https://doi.org/10.1090/S0002-9939-2014-12125-3
Keywords: Conformal invariant, inner radius, holomorphic function, Schwarzian derivative.
Received by editor(s): March 10, 2012
Received by editor(s) in revised form: September 10, 2012, September 27, 2012, and November 30, 2012
Published electronically: July 10, 2014
Additional Notes: The research of the first author was supported by the Russian Foundation for Basic Research, project 11-01-00038
The research of the second author was supported by the Academy of Finland, project 2600066611
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society