# Minkowski's Inequality for Integrals

## Theorem

Let $f, g$ be (Darboux) integrable functions.

Let $p \in \R$ such that $p > 1$.

Then:

$\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1/p} \le \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$

### Condition for Equality

$\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1/p} = \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$

holds if and only if, for all $x \in \closedint a b$:

$\dfrac {\map f x} {\map g x} = c$

for some $c \in \R_{>0}$.

## Proof

Define:

$q = \dfrac p {p - 1}$

Then:

$\dfrac 1 p + \dfrac 1 q = \dfrac 1 p + \dfrac {p - 1} p = 1$

It follows that:

 $\ds \int_a^b \size {\map f x + \map g x}^p \rd x$ $=$ $\ds \int_a^b \size {\map f x} \size {\map f x + \map g x}^{p - 1} \rd x + \int_a^b \size {\map g x} \size {\map f x + \map g x}^{p - 1} \rd x$ $\ds$ $\le$ $\ds \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} \paren {\int_a^b \paren {\size {\map f x + \map g x}^{p - 1} }^q \rd x}^{1 / q} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p} \paren {\int_a^b \paren {\size {\map f x + \map g x}^{p - 1} }^q \rd x}^{1 / q}$ Hölder's Inequality for Integrals (twice) $\ds$ $=$ $\ds \paren {\paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p} } \paren {\int_a^b \paren {\size {\map f x + \map g x}^{p - 1} }^q \rd x}^{1 / q}$ simplifying $\ds$ $=$ $\ds \paren {\paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p} } \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1 / q}$ Power of Power, and by hypothesis: $\paren {p - 1} q = p$ $\ds \leadsto \ \$ $\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1 - 1 / q}$ $\le$ $\ds \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$ dividing by $\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1 / q}$ $\ds \leadsto \ \$ $\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1 / p}$ $\le$ $\ds \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$ as $1 - \dfrac 1 q = p$

## Source of Name

This entry was named for Hermann Minkowski.