Maximin Polynomials and Inverse Balayage

Abstract

For a bounded domain G in the complex plane, we focus on the problem of maximizing the minimum on the boundary ∂G of (weighted) polynomials of degree n having all zeros in a set D ⊂ G. For arbitrary unit measures μ on ∂ G and weight w:= exp{Uμ}, the n-th root asymptotics of $$\matrix{{\rm sup} \cr pn} \matrix{{\rm inf} \cr z \varepsilon \partial G} \mid P_n(z)\omega^n(z) \mid$$ is considered and related to the existence and construction of an inverse balayage of μ on \(\overline D\), i.e. of a measure such that μ is its balayage when sweeping to ∂G.