An isoperimetric inequality for logarithmic capacity

We prove a sharp lower bound of the form cape E ≥ (1/2)diam E - ψ(area E/(1/4π diam² 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. Pólya's inequality cape ≥ (area E/π)^1/2. We give explicit formulations, functions of 1/2 diam E, for the extremal domains which we identify.