Hölder's Inequality for Sums

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

The Inequality

 * $$\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 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:


 * $$\alpha \beta \le \int_{0}^{\alpha} t^{p-1}dt + \int_{0}^{\beta} u^{q-1} du = \frac {\alpha^p} {p} + \frac {\beta^q} {q}$$

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:


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

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

Setting $$\alpha = \left|{a_k}\right|, \beta = \left|{b_k}\right|$$, the just established inequality tells us that:
 * $$\left|{a_k b_k}\right| \le \frac{1}{p} \left|{a_k}\right|^p + \frac{1}{q}\left|{b_k}\right|^q$$.

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

Now to prove Hölder's Inequality.

Let $$x$$ be in ℓp and $$y$$ in ℓq.

(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 $$a_k = \frac{x_k}{(\sum_{k=1}^\infty |x_k|^p)^{1/p}}$$ and $$b_k = \frac{y_k}{(\sum_{k=1}^\infty |y_k|^q)^{1/q}}$$.

Then clearly $$\sum \left|{a_k}\right|^p = \sum \left|{b_k}\right|^q = 1$$ and by our already established claim we have $$\sum \left|{a_k b_k}\right| \le 1$$.

Translating it back in terms of $$x$$ and $$y$$, and multiplying both sides by the denominator, we have:
 * $$\sum \limits_{k=1}^{\infty} \left|{x_k\,y_k}\right| \le \left({\sum_{k=1}^{\infty} \left|{x_k}\right|^p}\right)^{\!1/p\;} \left({\sum_{k=1}^{\infty} \left|{y_k}\right|^q}\right)^{\!1/q}$$.

Hence Hölder's Inequality is established.

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