  In this paper we show that the weighted Bernstein-Walsh inequality in
logarithmic potential theory is sharp up to some new universal constant,
provided that the external field is given by a logarithmic potential. Our main
tool for such results is a new technique of discretization of logarithmic
potentials, where we take the same starting point as in earlier work of Totik
and of Levin \& Lubinsky, but add an important new ingredient, namely some new
mean value property for the cumulative distribution function of the underlying
measure. As an application, we revisit the work of Beckermann \& Kuijlaars on
the superlinear convergence of conjugate gradients. These authors have
determined the asymptotic convergence factor for sequences of systems of linear
equations with an asymptotic eigenvalue distribution. There was some numerical
evidence to let conjecture that the integral mean of Green functions occurring
in their work should also allow to give inequalities for the rate of
convergence if one makes a suitable link between measures and the eigenvalues
of a single matrix of coefficients. We prove this conjecture , at least for a
class of measures which is of particular interest for applications.
