Inequalities of Levin-Ste\v{c}kin, Clausing and Chebyshev revisited

TL;DR

This paper provides elementary proofs of the Levin-Steckin and Clausing inequalities using Chebyshev's inequality and symmetrization techniques, simplifying their understanding and derivation.

Contribution

It introduces elementary proof methods for these inequalities, highlighting the role of symmetry and modified Chebyshev's inequality.

Findings

Elementary proof of Levin-Steckin inequality

Elementary proof of Clausing's inequality

Use of symmetrization and modified Chebyshev's inequality

Abstract

We prove the Levin-Ste\v{c}kin inequality using Chebyshev's inequality and symmetrization. Symmetry and slightly modified Chebyshev's inequality are also the key to an elementary proof of Clausing's inequality .