Hölder's Inequality (Special Case)

Introduction

Hölder's inequality is a fundamental inequality concerning Lebesgue spaces.

Hölder's Inequality helps in proving the Minkowski inequality which in turns helps in establishing that $\ell^p$ is a metric space under the metric defined by:

$\displaystyle d(x,y) = \left({\sum_k^\infty \left|{x_k - y_k}\right|^p}\right)^{1/p}$

The Inequality

$\displaystyle \sum \limits_{k=1}^{\infty} \left|{x_k\,y_k}\right| \le \left({\sum_{k=1}^{\infty} \left|{x_k}\right|^p}\right)^{1/p} \left({\sum_{k=1}^{\infty} |y_k|^q}\right)^{\!1/q} \text{ for all } \left({x_k}\right)_{k \in \N}, \left({y_k}\right)_{k \in \N} \in \C^\N$

Here $1 < p < \infty$ and $q$ is chosen such that $1/p + 1/q = 1$.

Proof

The blue colored region corresponds to $\displaystyle \int_0^\alpha t^{p-1} \mathrm d t$ and the red colored region to $\displaystyle \int_0^\beta u^{q-1} \mathrm d u$.

The proof of Hölder's inequality involves establishing an auxiliary inequality in the first stage and then using that auxiliary inequality to prove Hölder's inequality.

Let $p > 1$ and choose $q$ to be such that $1/p + 1/q = 1$.

Hence $(p-1)(q-1) = 1$ and so $1/(p-1) = q-1$.

Accordingly $u = t^{p-1}$ if and only if $t = u^{q-1}$.

Let $\alpha, \beta$ be any positive real numbers.

Since $\alpha \beta$ is the area of the rectangle in the given figure, we have:

$\displaystyle \alpha \beta \le \int_0^\alpha t^{p-1} \mathrm d t + \int_0^\beta u^{q-1} \mathrm d u = \frac {\alpha^p} p + \frac {\beta^q} q$

Note that even if the graph intersected the side of the rectangle corresponding to $t = \alpha$, this inequality would hold.

Also note that if either of $\alpha, \beta$ were zero then this inequality would hold trivially.

Now we turn our attention towards proving Hölder's Inequality.

We first establish a claim involving sequences $(a_n)$ and $(b_n)$ which have the property:

$\displaystyle \sum \left|{a_k}\right|^p = \sum \left|{b_k}\right|^q = 1$.

We claim that $\displaystyle \sum \left|{a_k b_k}\right| \le 1$.

Setting $\alpha = \left|{a_k}\right|, \beta = \left|{b_k}\right|$, the just established inequality tells us that:

$\displaystyle \left|{a_k b_k}\right| \le \frac 1 p \left|{a_k}\right|^p + \frac 1 q \left|{b_k}\right|^q$

Summing over all $k$ gives us $\displaystyle \sum \left|{a_k b_k}\right| \le \frac 1 p + \frac 1 q = 1$ which was our claim.

Now to prove Hölder's Inequality.

Let $x$ be in $\ell^p$ and $y$ in $\ell^q$.

(For other choices of $x$ and $y$ the RHS of the inequality is infinity and hence in those cases the inequality holds trivially.)

Also suppose that $x$ and $y$ are non zero, for otherwise the inequality is trivial.

Set:

$\displaystyle a_k = \frac{x_k}{\left({\sum \limits_{k=1}^\infty |x_k|^p}\right)^{1/p}}$

and

$\displaystyle b_k = \frac{y_k}{\left({\sum \limits_{k=1}^\infty |y_k|^q}\right)^{1/q}}$

Then clearly:

$\displaystyle \sum \left|{a_k}\right|^p = \sum \left|{b_k}\right|^q = 1$

and by our already established claim we have:

$\displaystyle \sum \left|{a_k b_k}\right| \le 1$

Translating it back in terms of $x$ and $y$, and multiplying both sides by the denominator, we have:

$\displaystyle \sum \limits_{k=1}^{\infty} \left|{x_k\,y_k}\right| \le \left({\sum_{k=1}^{\infty} \left|{x_k}\right|^p}\right)^{\!1/p\;} \left({\sum_{k=1}^{\infty} \left|{y_k}\right|^q}\right)^{\!1/q}$

Hence Hölder's Inequality is established.

$\blacksquare$

Source of Name

This entry was named for Otto Ludwig Hölder.

It was first found by L. J. Rogers in 1888, and discovered independently by Hölder in 1889.