Minkowski's Inequality

# Minkowski's Inequality

Recall from the Hölder's Inequality page that if $p, \geq 1$ and $q$ is the conjugate index of $p$ then for all measurable functions $f, g : X \to \mathbb{C}$ of a measure space $(X, \mathfrak T, \mu)$ we have that:

(1)
\begin{align} \quad \int |fg| \: d \mu \leq \| f \|_p \| g \|_q \end{align}

We use this to prove yet another important inequality called Minkowski's inequality.

 Theorem 1 (Minkowski's Inequality): Let $(X, \mathfrak T, \mu)$ be a measure space and let $1 \leq p \leq \infty$. Then for all $p$-integrable functions $f, g \in \mathcal L^p (X, \mathfrak T, \mu)$ we have that $\| f + g \|_p \leq \| f \|_p + \| g \|_p$.
• Proof: Let $f, g \in \mathcal L^p (X, \mathfrak T, \mu)$. Observe from the main theorem on the Lebesgue Spaces page, that for $1 \leq p \leq \infty$ we have that if $f, g \in \mathcal L^p(X, \mathfrak T, \mu)$ then $(f + g) \in \mathcal L^p (X, \mathfrak T, \mu)$. So we indeed may consider the $p$-norm of $f + g$.
• There are a few cases to consider.
• Case 1: Suppose that $p = 1$. Then by the triangle inequality for the absolutely value function we have that:
(2)
\begin{align} \quad \| f + g \|_1 = \left ( \int_X |f + g|^1 \: d \mu \right )^1 = \int_X |f + g| \: d \mu \leq \int_X (|f| + |g|) \: d \mu = \int_X |f| \: d \mu + \int_X |g| \: d \mu = \| f \|_1 + \| g \|_1 \end{align}
• Case 2: Suppose that $1 < p < \infty$. Note that if $\| f + g \|_p = 0$ then the inequality holds trivially. So we may assume that $\| f + g \|_p \neq 0$. Then by the triangle inequality we have that:
(3)
\begin{align} \quad \| f + g \|_p^p &= \left [ \left ( \int_X | f + g |^p \: d \mu \right )^{1/p} \right ]^p \\ &= \int_X |f + g|^p \: d \mu \\ &= \int_X |f + g| \cdot | f + g|^{p-1} \: d \mu \\ & \overset{\mathrm{Tri.}} \leq \int_X (|f| + |g|) \cdot |f + g|^{p-1} \: d \mu \\ & \leq \int_X [|f||f + g|^{p-1} + |g||f + g|^{p-1}] \: d \mu \\ & \leq \int_X |f||f + g|^{p-1} \: d \mu + \int_X |g||f + g|^{p-1} \: d \mu \end{align}
• We now estimate the following integrals:
(4)
\begin{align} \quad \int_X |f||f + g|^{p-1} \: d \mu \: (*) \quad , \quad \int_X |g||f + g|^{p-1} \: d \mu \: (**) \end{align}
• First observe that:
(5)
\begin{align} \quad \frac{\| f + g \|_p^p}{\| f + g \|_p} = \frac{\int_X |f + g|^p \: d \mu}{\left (\int_X |f + g|^p \: d \mu \right )^{1/p}} = \left ( \int_X |f + g|^p \: d \mu \right )^{1 - \frac{1}{p}} = \left ( \int_X |f + g|^p \: d \mu \right )^{\frac{p-1}{p}} \quad (\dagger) \end{align}
• For the integral $(*)$, by Hölder's Inequality we have that:
(6)
\begin{align} \quad \int_X |f||f + g|^{p-1} \: d \mu & \leq \| f \|_p \| (f + g)^{p-1} \|_q \\ & \leq \| f \|_p \| f + g \|_{\frac{p}{p-1}} \\ & \leq \| f \|_p \left ( \int_X (| f + g |^{(p-1)})^{\frac{p}{p-1}} \: d \mu \right )^{\frac{p-1}{p}} \\ & \leq \| f \|_p \left ( \int_X | f + g|^p \: d \mu \right )^{\frac{p-1}{p}} \\ & \overset{(\dagger)} \leq \| f \|_p \frac{\| f + g \|_p^p}{\| f + g\|_p} \end{align}
• For the integral $(**)$, by Hölder's Inequality we have that:
(7)
\begin{align} \quad \int_X |g||f + g|^{p-1} \: d \mu & \leq \| g \|_p \| (f + g)^{p-1} \|_q \\ & \leq \| g \|_p \| f + g \|_{\frac{p}{p-1}} \\ & \leq \| g \|_p \left ( \int_X (| f + g |^{(p-1)})^{\frac{p}{p-1}} \: d \mu \right )^{\frac{p-1}{p}} \\ & \leq \| g \|_p \left ( \int_X | f + g|^p \: d \mu \right )^{\frac{p-1}{p}} \\ & \overset{(\dagger)} \leq \| g \|_p \frac{\| f + g \|_p^p}{\| f + g\|_p} \end{align}
• Therefore we have that:
(8)
\begin{align} \quad \| f + g \|_p^p \leq (\| f \|_p + \| g \|_p) \frac{\| f + g \|_p^p}{\|f + g \|_p} \end{align}
• Multiply both sides of the inequality above by $\displaystyle{\frac{\| f + g\|_p}{\| f + g \|_p^p} > 0}$ to get:
(9)
\begin{align} \quad \| f + g \|_p \leq \| f \|_p + \| g \|_p \end{align}
• Case 3: Suppose that $p = \infty$. We note that if $|a + b| \leq M$ then $M$ need not be an upper bound for $|a| + |b|$. So:
(10)
\begin{align} \quad \| f + g \|_{\infty} &= \inf \{ M > 0 : |f(x) + g(x)| \leq M \: \mu - \mathrm{a.e. \: on} \: X \} \\ & \leq \inf \{ M > 0 : |f(x)| \leq M \: \mu - \mathrm{a.e. \: on} \: X \} + \inf \{ N > 0 : |g(x)| \leq N \: \mu - \mathrm{a.e. \: on} \: X \} \\ & \leq \| f \|_{\infty} + \| g \|_{\infty} \end{align}
• So Minkowski's inequality holds for all cases $1 \leq p \leq \infty$. $\blacksquare$