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

Abstract

Abstract. Let D be a convex planar domain of finite inradius R D . Fix the point 0 ∈ 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 θ of the Green's function G D (0, ρe iθ ), for fixed ρ, 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.

Old and New on the Bass Note, the Torsion Function and the Hyperbolic Metric

“…Precise results on rearrangements of the Green's function of a convex domain when the pole of the Green's function lies at the centre of the largest disk in the domain were obtained by Bañuelos, Carroll and Housworth [3].…”

Planar Brownian Motion and Complex Analysis

“…Interesting connections have been found between planar Brownian motion and the hyperbolic metric on simply connected domains in [2,5,8].…”

Error estimates for harmonic and biharmonic interpolation splines with annular geometry

“…We leave the question open whether in Theorem 3 one may replace the L 2 -norm by the supremum norm. In passing we mention that in [39] the inequality (9) has been generalized to L-splines where L is a differential operator with constant coefficients of order 4.…”

“…the fundamental work of G. Pólya, G. Szegö about isoperimetric inequalities in [44], or the monography [51]. There is a vast literature on this subject with many ramifications and it would take too much space to survey the results, so we only mention a very incomplete list of new references [9], [15], [16], [17], [18]. In Section 2 we shall provide a self-contained proof of Theorem 4 which is based on a Green function approach.…”