## Hölder’s inequality via complex analysis

In this post I will give a complex variables proof of Hölder’s inequality due to Rubel. The argument is very similar to Thorin’s proof of the Riesz-Thorin interpolation theorem. I imagine that there is a multilinear form of Riesz-Thorin that provides a common generalization of the two arguments, however we won’t explore this here. We start by establishing the well-known three lines lemma.

**Lemma **(three lines lemma) Let be a bounded analytic function in the strip . Furthermore, assume that extends to a continuous function on the boundary of and satisfies

for . Then, for , we have that

Proof: Let and consider the function (analytic in )

One easily checks that if or . Furthermore, since is uniformly bounded for , we must have that

for . We now claim that, for sufficiently large, for where . This follows by the previous remarks on the boundary of , and by the maximum modulus principle in its interior. This completes the proof.

We are now ready to give a complex variables proof of Hölder’s Inequality.

**Theorem** (Hölder’s Inequality)Let be a measure space, such that . If and then

Proof: By a standard limiting argument (preformed first with, say, fixed) it will suffice to assume that and are simple functions. If we let we may now rewrite Hölder’s inequality as

Indeed, . Using the fact that and are simple, we can define a function, , analytic in the strip by

It follows that , and that is bounded on the closure of . We record that and . Now, by the three lines lemma, we have that

Taking we recover Hölder’s inequality.

Updated 10/13/2009: typos corrected

Updated 10/14/2009: the statement of the three lines lemma was truncated in the original post

Update 10/31/2009: typo in definition of B.

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