We prove a sharp lower bound of the form cap E ≥ (1/2)diam E ·Ψ(area E/((π/4) 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 ≥ (1/4)diam E and G. Pòlya's inequality cap E ≥ (area E/π)1/2. We give explicit formulations, functions of (1/2)diam E, for the extremal domains which we identify.
|Journal||Annales Academiae Scientiarium Fennicae Mathematica|
|State||Published - 2002|