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Proceedings of the American Mathematical Society

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

 

 

Inradius and integral means
for green's functions and conformal mappings


Authors: Rodrigo Bañuelos, Tom Carroll and Elizabeth Housworth
Journal: Proc. Amer. Math. Soc. 126 (1998), 577-585
MSC (1991): Primary 30C45
MathSciNet review: 1443813
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Abstract: Let $D$ be a convex planar domain of finite inradius $R_D$. Fix the point $0\in D$ and suppose the disk centered at $0$ and radius $R_D$ is contained in $D$. Under these assumptions we prove that the symmetric decreasing rearrangement in $ \theta $ of the Green's function $G_{D}(0, \rho e^{i\theta})$, for fixed $\rho$, is dominated by the corresponding quantity for the strip of width $2R_D$. From this, sharp integral mean inequalities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of $D$ with that of the strip and a careful analysis of geodesics.


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Additional Information

Rodrigo Bañuelos
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: banuelos@math.purdue.edu

Tom Carroll
Affiliation: Department of Mathematics, University College, Cork, Ireland
Email: tc@ucc.ie

Elizabeth Housworth
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: eah@math.uoregon.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04217-8
Keywords: Integral means, Green's functions, symmetric decreasing rearrangements, Baernstein star functions
Received by editor(s): August 23, 1996
Additional Notes: Research of the first author supported in part by NSF under grant DMS9400854, of the second author by the President’s Research Fund, University College, Cork, and of the third author by NSF under grant DMS9501611.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society