INRADIUS AND INTEGRAL MEANS FOR GREENS-FUNCTIONS AND CONFORMAL-MAPPINGS

Citation
R. Banuelos et al., INRADIUS AND INTEGRAL MEANS FOR GREENS-FUNCTIONS AND CONFORMAL-MAPPINGS, Proceedings of the American Mathematical Society, 126(2), 1998, pp. 577-585
Citations number
9
Categorie Soggetti
Mathematics,Mathematics,Mathematics,Mathematics
ISSN journal
00029939
Volume
126
Issue
2
Year of publication
1998
Pages
577 - 585
Database
ISI
SICI code
0002-9939(1998)126:2<577:IAIMFG>2.0.ZU;2-5
Abstract
Let D be a convex planar domain of finite inradius R-D. Fix the point 0 is an element of D and suppose the disk centered at 0 and radius R-D is contained in D. Under these assumptions we prove that the symmetri c decreasing rearrangement in theta of the Green's function G(D)(0, rh o e(i theta)), for fixed rho, is dominated by the corresponding quanti ty for the strip of width 2R(D). From this, sharp integral mean inequa lities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison esti mates for the hyperbolic metric of D with that of the strip and a care ful analysis of geodesics.