The extremal function for the variational 2-capacity of a compact subset of the Royden harmonic boundary of an open Riemann surface relative to an end of , referred to as the capacitary function of , is characterized as the Dirichlet finite harmonic function on vanishing continuously on the relative boundary of satisfying the following three properties: the normal derivative measure of exists on with on ; on ; quasieverywhere on . As a simple application of the above characterization, we will show the validity of the following inequality
for every compact subset of , where is the harmonic measure of calculated at a fixed point in and is a constant depending only upon the triple .
Mitsuru NAKAI. "Extremal functions for capacities." J. Math. Soc. Japan 61 (2) 345 - 361, April, 2009. https://doi.org/10.2969/jmsj/06120345