Post History

Using convexity in the proof of Hölder’s inequality

A key fact for the algebra properties of $L^p$ spaces is [Hölder’s inequality](https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality):

> Let $f \in L^p$ and $g \in L^q$ for some $0 < p,q \leq \infty$. Then $fg \in L^r$ and $\|fg\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^q}$, where the exponent $r$ is defined by the formula $\frac{1}{r} = \frac{1}{p} + \frac{1}{q}$.

There are many known proofs for this fact, including one from the Wikipedia link above. This post focuses on the specific proof in Tao's notes. 

The endpoint cases where $p=\infty$ or $q=\infty$ can be proved separately and easily, usually left as exercises for dealing with the $L^\infty$ norm. 


In his real analysis lecture [note][1], Terry Tao reduces the statement, using homogeneity, to the nontrivial essential cases where $p,q<\infty$, $r=1$, and  $\|f\|_{L^p}=\|g\|_{L^q} = 1$. I very much like how he isolates the essential component of the proof by reducing the statement to simpler cases, which is conceptually (and pedagogically) natural to understand and is a problem technique used often. 


The following is his proof:


>Our task is now to show that
$$\int_X |fg|\ d\mu \leq 1. \tag{1}$$
>Here, we use the convexity of the exponential function $t \mapsto e^t$ on ${}[0,+\infty)$, which implies the convexity of the function $t \mapsto |f(x)|^{p(1-t)} |g(x)|^{qt}$ for $t \in [0,1]$ for any $x$. In particular, we have
$$
|f(x) g(x)| \leq \frac{1}{p} |f(x)|^p + \frac{1}{q} |g(x)|^q \tag{2}
$$
and the claim $(1)$ follows from the normalisations on $p, q, f, g.$

**Questions:**

 - How does the convexity of the exponential function imply the convexity of the function $t \mapsto |f(x)|^{p(1-t)} |g(x)|^{qt}$ for $t \in [0,1]$ for any $x$?
 - How does one get (2)?

---
**Notes.** Answers to these questions are rather trivial to experienced readers but may require some effort for beginners. I will write my own answers below. 





  [1]: https://terrytao.wordpress.com/2009/01/09/245b-notes-3-lp-spaces/