# Minkowski's Inequality for Sums/Index Greater than 1

## Theorem

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers.

Let $p \in \R$ be a real number such that $p > 1$.

Then:

$\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / p} \le \paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p}$

## Proof 1

Define:

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

Then:

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

It follows that:

 $\ds \sum_{k \mathop = 1}^n \paren {a_k + b_k}^p$ $=$ $\ds \sum_{k \mathop = 1}^n a_k \paren {a_k + b_k}^{p - 1} + \sum_{k \mathop = 1}^n b_k \paren {a_k + b_k}^{p - 1}$ $\ds$ $\le$ $\ds \paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \paren {\paren {a_k + b_k}^{p - 1} }^q}^{1 / q} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \paren {\paren {a_k + b_k}^{p - 1} }^q}^{1 / q}$ Hölder's Inequality for Sums (twice) $\ds$ $=$ $\ds \paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / q} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / q}$ Power of Power, and by hypothesis: $\paren {p - 1} q = p$ $\ds$ $=$ $\ds \paren {\paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p} } \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / q}$ $\ds \leadsto \ \$ $\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 - 1 / q}$ $\le$ $\ds \paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p}$ dividing by $\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / q}$ $\ds \leadsto \ \$ $\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / p}$ $\le$ $\ds \paren {\sum_{k \mathop = 1}^n {a_k}^p}^{1 / p} + \paren {\sum_{k \mathop = 1}^n {b_k}^p}^{1 / p}$ as $1 - \dfrac 1 q = p$

## Proof 2

Let $\mathbf a$ and $\mathbf b$ be real finite sequences:

 $\ds \mathbf a$ $=$ $\ds \sequence {a_k}_{1 \mathop \le k \mathop \le n}$ $\ds \mathbf b$ $=$ $\ds \sequence {b_k}_{1 \mathop \le k \mathop \le n}$

Let $\norm {\mathbf a}_p$ denote the $p$-norm of $\mathbf a$:

$\norm {\mathbf a}_p := \paren {\ds \sum_{k \mathop = 1}^n {a_k}^p}^{1 / p}$

Define:

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

Then:

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

Then:

 $\ds \paren {\norm {\mathbf a + \mathbf b}_p}^p$ $=$ $\ds \norm {\paren {\mathbf a + \mathbf b}^p}_1$ $\ds$ $=$ $\ds \norm {\mathbf a \paren {\mathbf a + \mathbf b}^{p - 1} + \mathbf b \paren {\mathbf a + \mathbf b}^{p - 1} }_1$ $\ds$ $=$ $\ds \norm {\mathbf a \paren {\mathbf a + \mathbf b}^{p - 1} }_1 + \norm {\mathbf b \paren {\mathbf a + \mathbf b}^{p - 1} }_1$ $\ds$ $\le$ $\ds \norm {\mathbf a}_p \norm {\paren {\mathbf a + \mathbf b}^{p - 1} }_q + \norm {\mathbf b}_p \norm {\paren {\mathbf a + \mathbf b}^{p - 1} }_q$ Hölder's Inequality for Sums (twice) $\ds$ $=$ $\ds \paren {\norm {\mathbf a}_p + \norm {\mathbf b}_p} \norm {\paren {\mathbf a + \mathbf b}^{p - 1} }_q$ simplifying $\ds$ $=$ $\ds \paren {\norm {\mathbf a}_p + \norm {\mathbf b}_p} \norm {\paren {\mathbf a + \mathbf b}^{p - 1} }_{p / \paren {p - 1} }$ by hypothesis: $\dfrac p {p - 1} = q$ $\ds$ $=$ $\ds \paren {\norm {\mathbf a}_p + \norm {\mathbf b}_p} \paren {\norm {\mathbf a + \mathbf b}_p}^{p - 1}$ Transformation of P-Norm $\ds \leadsto \ \$ $\ds \norm {\mathbf a + \mathbf b}_p$ $\le$ $\ds \norm {\mathbf a}_p + \norm {\mathbf b}_p$ dividing both sides by $\paren {\norm {\mathbf a + \mathbf b}_p}^{p - 1}$

$\blacksquare$

## Source of Name

This entry was named for Hermann Minkowski.