The Isoperimetric Inequality and Wirtinger’s Inequality

Author

• Mason Neal, University of Mississippi Main CampusFollow

Date of Award

Spring 5-7-2026

Document Type

Undergraduate Thesis

Department

Mathematics

First Advisor

Micah Milinovich

Second Advisor

Laura Sheppardson

Third Advisor

Leo Stein

Relational Format

Dissertation/Thesis

Abstract

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

Recommended Citation

Neal, Mason, "The Isoperimetric Inequality and Wirtinger’s Inequality" (2026). Honors Theses. 3477.
https://egrove.olemiss.edu/hon_thesis/3477