On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions

TL;DR

This paper studies functions with specific mean value properties related to midpoints and weighted averages, revealing connections to harmonic functions and exploring multivariable analogues and weighted harmonic functions.

Contribution

It introduces a class of functions characterized by weighted mean value properties and explores their links to harmonic functions and multivariable generalizations.

Findings

Functions with midpoint mean value property are characterized.

Connections established between these functions and harmonic functions.

Exploration of multivariable analogues and weighted harmonic functions.

Abstract

We investigate functions with the property that for every interval, the slope at the midpoint of the interval is the same as the average slope. More generally, we find functions whose average slopes over intervals are given by the slope at a weighted average of the endpoints of those intervals. This is equivalent to finding functions satisfying a weighted mean value property. In the course of our exploration, we find connections to harmonic functions that prompt us to explore multivariable analogues and the existence of "weighted harmonic functions."