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

DOI:
https://doi.org/10.1090/S0002-9939-98-04217-8

MathSciNet review:
1443813

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a convex planar domain of finite inradius . Fix the point and suppose the disk centered at and radius is contained in . Under these assumptions we prove that the symmetric decreasing rearrangement in of the Green's function , for fixed , is dominated by the corresponding quantity for the strip of width . 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 with that of the strip and a careful analysis of geodesics.

**1.**C. Bandle,*Isoperimetric Inequalities and Applications*, Pitman, London, 1980. MR**81e:35095****2.**R. Bañuelos and T. Carroll, ``Brownian Motion and the Fundamental Frequency of a Drum'',*Duke Math. J.***75**(1994) 575-602. MR**96m:31003****3.**R. Bañuelos and E. Housworth, ``An Isoperimetric-Type Inequality for Integrals of Green's Functions'',*Michigan Math. J.***42**(1995) 603-611. MR**96j:30038****4.**P. Duren,*Univalent Functions*, Springer-Verlag, New York, 1983. MR**85j:30034****5.**D. Girela, ``Inequalities for the Integral Means of BMOA Functions'',*Complex Variables Theory Appl.***25**(1994) 43-48. MR**95m:30046****6.**C. D. Minda, ``Lower Bounds for the Hyperbolic Metric in Convex Regions'',*Rocky Mountain J. Math***13**(1983) 61-69. MR**84j:30039****7.**M. Nowak, ``Some Inequalities for BMOA Functions, ''*Complex Variables Theory Appl.***16**(1991) 81-86. MR**92e:30019****8.**Ch. Pommerenke,*Boundary Behavior of Conformal Maps*, Springer-Verlag, New York, 1992. MR**95b:30008****9.**R. Sperb,*Maximum Principles and Their Applications*, Academic Press, New York, 1981. MR**84a:35033**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
30C45

Retrieve articles in all journals with MSC (1991): 30C45

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:
https://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