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

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.