# The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

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## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Weighted AM–GM Inequality.**If $0\le {c}_{i}\in \mathbb{R}\phantom{\rule{3.33333pt}{0ex}}(i=1,\dots ,n)$ and $0\le {\lambda}_{i}\in \mathbb{R}\phantom{\rule{3.33333pt}{0ex}}(i=1,\dots ,n)$ such that $\sum _{i=1}^{n}}{\lambda}_{i}=1$, then

**Hölder Inequality.**If $0\le {a}_{i},{b}_{i}\in \mathbb{R}\phantom{\rule{3.33333pt}{0ex}}(i=1,\dots ,n)$ and $p,q\in {\mathbb{R}}^{+}$ such that ${p}^{-1}+{q}^{-1}$ = 1, then

**Weighted Power-Mean Inequality.**If $0\le {c}_{i},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{i}\in \mathbb{R}\phantom{\rule{3.33333pt}{0ex}}(i=1,\dots ,n)$ such that $\sum _{i=1}^{n}}{\lambda}_{i}=1$, and $r,s\in {\mathbb{R}}^{+}$ such that $r\le s$, then

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Theorem**

**3.**

**Proof.**

## 3. Concluding Remarks

- Equivalence of the Hölder’s inequality and the Minkowski inequality; see [9].
- Equivalence of the Cauchy–Schwarz inequality and the Hölder’s inequality; see [8].
- Equivalence of the Cauchy–Schwarz inequality and the Covariance–Variance inequality; see [7].
- Equivalence of the Kantorovich inequality and the Wielandt inequality; see e.g., [11]
- Equivalence of the AM–GM inequality and the Bernoulli inequality; see e.g., [10]
- Equivalence of the Hölder inequality and Artin’s theorem; see e.g., [12] (pp. 657–663) for details.
- Equivalence of the Hölder inequality and the weighted AM–GM inequality; refer to Theorem 1.
- Equivalence of the Hölder inequality and the weighted power-mean inequality; see Theorem 2.
- Equivalence of the weighted power-mean inequality and the weighted AM–GM inequality; refer to Theorem 3.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AM–GM | Arithmetic Mean–Geometric Mean |

CBS | Cauchy–Bunyakovsky–Schwarz |

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**MDPI and ACS Style**

Li, Y.; Gu, X.-M.; Zhao, J.
The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality. *Symmetry* **2018**, *10*, 380.
https://doi.org/10.3390/sym10090380

**AMA Style**

Li Y, Gu X-M, Zhao J.
The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality. *Symmetry*. 2018; 10(9):380.
https://doi.org/10.3390/sym10090380

**Chicago/Turabian Style**

Li, Yongtao, Xian-Ming Gu, and Jianxing Zhao.
2018. "The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality" *Symmetry* 10, no. 9: 380.
https://doi.org/10.3390/sym10090380