Inradius and integral means for Green’s functions and conformal mappings

Abstract:

Let $D$ be a convex planar domain of finite inradius $R_D$. Fix the point $0\in D$ and suppose the disk centered at $0$ and radius $R_D$ is contained in $D$. Under these assumptions we prove that the symmetric decreasing rearrangement in $\theta$ of the Green’s function $G_{D}(0, \rho e^{i\theta })$, for fixed $\rho$, is dominated by the corresponding quantity for the strip of width $2R_D$. 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.