The Weighted Arithmetic Mean-Geometric Mean Inequality is Equivalent to the H\"{o}lder Inequality

TL;DR

This paper establishes the mathematical equivalence among the weighted AM-GM inequality, the H"{o}lder inequality, and the weighted power-mean inequality, providing generalized proofs that unify these fundamental inequalities.

Contribution

It proves the equivalence of these inequalities and extends previous results to more generalized forms.

Findings

Proves the equivalence of weighted AM-GM, H"{o}lder, and power-mean inequalities.

Provides fully detailed proofs of the mathematical equivalence.

Generalizes previous results to broader inequality forms.

Abstract

In the current note, we investigate the mathematical relations among the weighted arithmetic mean-geometric mean (AM-GM) inequality, the H\"{o}lder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM-GM inequality, the weighted power-mean inequality and the H\"{o}lder inequality are fully achieved. The new results are more generalized than those of previous studies.