# Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure space with the folowing method:

differentiate $p\to \log \| f\|_{\frac1{p}}$ twice and observe that the result is positive.

Remark: the inequality is equivalent to the Hölder inequality.

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Perform the differentiation: \begin{align} \frac{\mathrm{d}^2}{\mathrm{d}p^2}\log\left(\int|f(x)|^p\;\mathrm{d}x\right) &=\frac{\mathrm{d}}{\mathrm{d}p}\frac{1}{\int|f(x)|^p\;\mathrm{d}x}\int\log(|f(x)|)\;|f(x)|^p\;\mathrm{d}x\\ &=-\frac{1}{\left(\int|f(x)|^p\;\mathrm{d}x\right)^2}\left(\int\log(|f(x)|)\;|f(x)|^p\;\mathrm{d}x\right)^2\\ &\phantom{=}+\frac{1}{\int|f(x)|^p\;\mathrm{d}x}\int\log(|f(x)|)^2\;|f(x)|^p\;\mathrm{d}x\tag{1} \end{align} and Jensen's inequality says that because $x\mapsto x^2$ is convex, $$\frac{1}{\int|f(x)|^p\;\mathrm{d}x}\int\log(|f(x)|)^2\;|f(x)|^p\;\mathrm{d}x\ge\left(\frac{1}{\int|f(x)|^p\;\mathrm{d}x}\int\log(|f(x)|)\;|f(x)|^p\;\mathrm{d}x\right)^2\tag{2}$$ Equations $(1)$ and $(2)$ show that $$\frac{\mathrm{d}^2}{\mathrm{d}p^2}\log\left(\int|f(x)|^p\;\mathrm{d}x\right)\ge0$$ thus, $p\mapsto\log\left(\int|f(x)|^p\;\mathrm{d}x\right)$ is a convex function. Therefore, since $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$, we get $$\frac{1}{r}\log\left(\int|f(x)|^r\;\mathrm{d}x\right)\le\frac{1-\theta}{p}\log\left(\int|f(x)|^p\;\mathrm{d}x\right)+\frac{\theta}{q}\log\left(\int|f(x)|^q\;\mathrm{d}x\right)$$ which becomes $$\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$$
What exactly are the hypotheses you're assuming here? Why is $|f|^p \cdot \log|f|$ integrable, why exactly are all your differentiations and exchanges of differentiation and integration justified? There seem quite a few details missing. –  t.b. Aug 22 '11 at 20:43
@Theo Buehler: In the differentiation, I am assuming that $f\in L^q$ for $q$ in some neighborhood of $p$. That being the case, then $log(|f|)\;|f|^p$ and $log(|f|)^2\;|f|^p$ are both integrable. All the details could be included here, but then the main idea that was being demonstrated would be obfuscated. Those details would be better left to the reader. –  robjohn Aug 22 '11 at 21:16