A Geometric Generalization of the Pythagorean Means

Summary

Each of the Pythagorean means corresponds to the centroid of a region in the Cartesian plane. We show how this insight leads to a short proof of a result that generalizes the HM-GM-AM inequality.

MSC::

• Primary 26E60
• Secondary 26D99

Acknowledgment

The authors would like to thank an anonymous referee for valuable comments that have improved the paper.

Notes

1 It is, of course, possible to give a definition of convexity that does not mention differentiability, but this is not required for our purposes.

2 This is permissible by the Leibniz integral rule, the preconditions of which are readily met as $g(x,t)=s^t$ is continuously differentiable. See Conrad [2] for an excellent exposition on this topic.

Additional information

Notes on contributors

Greg Markowsky

GREG MARKOWSKY obtained his Ph.D. at the City University of New York (CUNY) and is now a lecturer at Monash University. His research interests include complex analysis, probability, graph theory, and good old calculus, as is used in this paper.

Dylan Phung

DYLAN PHUNG is a student at Yarra Valley Grammar School.

David Treeby

DAVID TREEBY is a research associate at Monash University, where he obtained his Ph.D. He enjoys digging for gems along the boundary of classical physics and mathematics. He also teached at Scotch College, Melbourne.