# Minkowski's Inequality for Integrals

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## Contents

## Theorem

Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$.

- $(1):\quad$ Let $p > 1$. Then:
- $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \le \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$

- $(2):\quad$ Let $p < 1, p \ne 0$. Then:
- $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \ge \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$

## Proof

## Source of Name

This entry was named for Hermann Minkowski.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 36$: Inequalities: $36.15$