An Isoperimetric Inequality for Logarithmic Capacity

Roger Barnard, Kent Pearce, Alexander Solynin

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)419-436
JournalAnnales Academiae Scientiarium Fennicae Mathematica
StatePublished - 2002


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