Reverse Triangle Inequalities for Potentials

TL;DR

This paper establishes sharp reverse inequalities for logarithmic potentials on compact sets, with applications to polynomial norms and external fields, advancing potential theory and polynomial approximation techniques.

Contribution

It introduces sharp additive constants for reverse inequalities of potentials and applies these results to weighted polynomial norms and external field scenarios.

Findings

Sharp additive constants for potential inequalities

Applications to generalized polynomial norms

New Riesz decomposition for weighted distance functions

Abstract

We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials $w^nP_n, deg(P_n)\le n,$ and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.