Minkowski's Inequality for Integrals

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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 $x \in \closedint a b$:

$\map g x = c \map f x$

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.


Sources

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