Use of logarithmic sums to estimate polynomials

Given any simple closed carve Gamma with enough smoothness we take any large number, say N, of points zeta (n) on Gamma, equispaced thereon with respect to harmonic measure, seen from infinity, for the exterior of Gamma. Then, if P is any polynomial of degree M < N, the values \P(z)\ can, for z inside Gamma, be estimated in terms of the logarithmic average (1/N) Sigma (N)(n=1) log(+) \P(zeta (n))\. When M and N both tend to oo the estimate holds uniformly for each fixed z inside Gamma as long as the ratio M/N remains bounded away from 1, and that requirement cannot be lightened. The least superharmonic majorant and its properties play an important role in the proof of this result; other tools used are Jensen's formula and the Koebe 1/4-theorem.