A note on weighted homogeneous Siciak-Zaharyuta extremal functions

TL;DR

This paper establishes a representation of weighted homogeneous Siciak-Zaharyuta extremal functions as envelopes of disc functionals for certain upper semicontinuous functions on complex cones, extending understanding of extremal functions in complex analysis.

Contribution

It proves that weighted homogeneous Siciak-Zaharyuta extremal functions can be expressed as envelopes of disc functionals under specific connectivity conditions.

Findings

Representation as envelope of disc functional

Applicable to upper semicontinuous functions on complex cones

Extends extremal function theory in complex analysis

Abstract

We prove that for any given upper semicontinuous function $\varphi$ on an open subset $E$ of $\mathbb C^n\setminus\{0\}$, such that the complex cone generated by $E$ minus the origin is connected, the homogeneous Siciak-Zaharyuta function with the weight $\varphi$ on $E$, can be represented as an envelope of a disc functional.