On functions whose mean value abscissas are midpoints, with connections to harmonic functions

Paul Carter; David Lowry-Duda

doi:10.4169/amer.math.monthly.124.6.535

On functions whose mean value abscissas are midpoints, with connections to harmonic functions

Paul Carter, David Lowry-Duda

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 analogs and the existence of "weighted harmonic functions".

Original language	English (US)
Pages (from-to)	535-542
Number of pages	8
Journal	American Mathematical Monthly
Volume	124
Issue number	6
DOIs	https://doi.org/10.4169/amer.math.monthly.124.6.535
State	Published - Jun 1 2017
Externally published	Yes

Bibliographical note

Funding Information:
ACKNOWLEDGMENTS. D. L.-D. was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 0228243. P. C. gratefully acknowledges support by the National Science Foundation through grant DMS-1148284.

Publisher Copyright:
© The Mathematical Association of America.

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N2 - 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 analogs and the existence of "weighted harmonic functions".

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On functions whose mean value abscissas are midpoints, with connections to harmonic functions

Abstract

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