Modified logarithmic potential theory and applications

TL;DR

This paper extends potential theory to analyze functions combining polynomials and holomorphic functions, providing estimates that facilitate probabilistic results like large deviation principles.

Contribution

It introduces a Bernstein-Walsh type estimate for composite functions of polynomials and holomorphic functions, advancing potential theory applications.

Findings

Derived a Bernstein-Walsh type estimate for specific function classes

Applied estimates to establish large deviation principles in probability

Enhanced understanding of potential theory in probabilistic contexts

Abstract

We develop potential theory including a Bernstein-Walsh type estimate for functions of the form $p(z)q(f(z))$ where $p,q$ are polynomials and $f$ is holomorphic. Such functions arise in the study of certain ensembles of probability measures and our estimates lead to probabilistic results such as large deviation principles.