Geometric mean of conformal radii for compact sets contained in a simply-connected domain

Neranjan Marasinghe

University of Mississippi

Abstract

The well-known results by G. Pélya and G. Szegö assert that the disk has the largest conformal radius among all simply connected domains of a fixed area and, the equilateral triangle and square have the largest conformal radii among all triangles with a given area and among all quadrilaterals with a given area, respectively.

In this talk, Dr. Neranjan Marasinghe will discuss three theorems that extend these results to geometric mean of conformal radii over compact subsets with prescribed positive area contained in a simply connected domain of a given area, in a triangle of a given area, and in a quadrilateral of a given area, respectively.

In addition to that, the geometric mean of conformal radii of rectifiable arcs contained in a simply connected domain will be defined and few related inequalities will be introduced. Few open problems related to this work will also be discussed.