Inradius and Integral Means for Green’s Functions and Conformal Mappings
نویسندگان
چکیده
Let D be a convex planar domain of finite inradius RD . Fix the point 0 ∈ D and suppose the disk centered at 0 and radius RD is contained in D. Under these assumptions we prove that the symmetric decreasing rearrangement in θ of the Green’s function GD(0, ρe iθ), for fixed ρ, is dominated by the corresponding quantity for the strip of width 2RD . 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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