# Introduction

I have seen Hölder’s inequality and Minkowski’s inequality proved in several ways but this seems the most perspicuous (to me at any rate).

# Young’s Inequality

If $a, b \ge 0$ and $p,q \ge 1$ such that

$\displaystyle \frac{1}{p} + \frac{1}{q} = 1$

then

$\displaystyle ab \le \frac{a^p}{p} + \frac{b^q}{q}$

A $p$ and $q$ satisfying the premise are known as conjugate indices.

Proof

Since $\log$ is convex we have

$\displaystyle t\log{x} + (1 - t)\log{y} \le \log{(tx + (1 - t)y)}$

Substituting in appropriate values gives

$\displaystyle \frac{1}{p}\log{a^p} + \frac{1}{q}\log{b^q} \le \log{\bigg(\frac{a^p}{p} + \frac{b^q}{q}\bigg)}$

or

$\displaystyle \log{a} + \log{b} \le \log{\bigg(\frac{a^p}{p} + \frac{b^q}{q}\bigg)}$

Now take exponents.

$\blacksquare$

# Hölders’s Inequality

Let $p$ and $q$ be conjugate indices with $1 < p < \infty$ and let $f \in L^p(\Omega)$ and $g \in L^q(\Omega)$ then $fg \in L^1(\Omega)$ and

$\displaystyle \|fg\|_{L^1} \le \|f\|_{L^p}\|g\|_{L^q}$

Proof

By Young’s inequality

$\displaystyle \int_\Omega \frac{|f(x)|}{\|f\|_{L^p}} \frac{|g(x)|}{\|g\|_{L^q}} \le \int_\Omega \frac{1}{p}\frac{|f(x)|^p}{\|f\|_{L^p}^p} + \frac{1}{q}\frac{|g(x)|^q}{\|g\|_{L^q}^q} = \frac{1}{p} + \frac{1}{q} = 1$

$\blacksquare$

By applying a counting measure to $\Omega$ we also obtain

$\displaystyle \sum |x_i y_i| \le \big(\sum |x_i|^p\big)^{1/p} \big(\sum |y_i|^q\big)^{1/q}$

# Minkowski’s Inequality

$\displaystyle \|f + g\|_{L^p} \le \|f\|_{L^p} + \|g\|_{L^p}$

Proof

By Hölder’s inequality

$\displaystyle \int_\Omega |f + g|^p \le \int_\Omega |f||f + g|^{p-1} + \int_\Omega |g||f + g|^{p-1} \le \|f\|_{L^p}A + \|g\|_{L^p}A$

where

$\displaystyle A = \||f + g|^{p-1}\|_{L^q} = \big(\int_\Omega |f(x) + g(x)|^p\big)^{1/q}$

and $A$ is finite since $L^p$ is a vector space.

$\blacksquare$