An isoperimetric inequality for logarithmic capacity

Abstract

We prove a sharp lower bound of the form cap E ≥ (½) diam E . Ψ(area E/(1/4π diam 2 E)) for the logarithmic capacity of a compact connected planar set E in terms of its area and diameter. Our lower bound includes as special cases G. Faber's inequality cap E > diam 1/4 E and G. POlya's inequality cap E > (area E/π) 1 / 2 . We give explicit formulations, functions of ½ diam E, for the extremal domains which we identify.