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

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Minkowski's Inequality for Integrals: $3.2.13$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: Minkowski's Inequality for Integrals: $36.15$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 37$: Inequalities: Minkowski's Inequality for Integrals: $37.15.$