# Refinements of the Converse Hölder and Minkowski Inequalities

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- L1:
- $f,g\in L\Rightarrow \left(\right)open="("\; close=")">\alpha f+\beta g$ for all $\phantom{\rule{4pt}{0ex}}\alpha ,\beta \in \mathbb{R}$;
- L2:
- $1\in L$, i.e., if $\phantom{\rule{4pt}{0ex}}f\left(t\right)=1$ for $t\in E$, then $\phantom{\rule{4pt}{0ex}}f\in L$.

**Theorem**

**1**

## 2. Refinement of the Converse Hölder Inequality

**Theorem**

**2.**

**Proposition**

**1**

**Theorem**

**3.**

**Proof.**

## 3. Refinement of the Converse Beckenbach Inequality

**Theorem**

**4**

**Theorem**

**5.**

**Proof.**

## 4. The Converse Minkowski Inequality and Its Refinements

**Theorem**

**6**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8**

**Proof.**

## 5. Applications on Mixed Means

**Theorem**

**9.**

**Corollary**

**1.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Pečarić, J.; Perić, J.; Varošanec, S.
Refinements of the Converse Hölder and Minkowski Inequalities. *Mathematics* **2022**, *10*, 202.
https://doi.org/10.3390/math10020202

**AMA Style**

Pečarić J, Perić J, Varošanec S.
Refinements of the Converse Hölder and Minkowski Inequalities. *Mathematics*. 2022; 10(2):202.
https://doi.org/10.3390/math10020202

**Chicago/Turabian Style**

Pečarić, Josip, Jurica Perić, and Sanja Varošanec.
2022. "Refinements of the Converse Hölder and Minkowski Inequalities" *Mathematics* 10, no. 2: 202.
https://doi.org/10.3390/math10020202