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
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.