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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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1 Answer 1

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 $$ \|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta $$

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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
    
Sure, I have no doubts about that (and your ability to do it). But that's my major complaint about this approach of proving the desired inequality. The formal argument is pretty simple, as you show aptly, but filling in the details looks rather messy to me. I'd rather go for the usual approach via AM-GM or Young which seems much clearer to me (and if one wants, one can prove these by using concavity of log and differentiation). (Edit @all there was an earlier comment to the end that the details can be filled in but that this would be obfuscating the argument). –  t.b. Aug 22 '11 at 21:03
    
@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
    
@Theo Buehler: sorry about pulling the comment temporarily, but I was not finished, yet I hit the return to add a paragraph, which is not allowed in the comments. As for your objections about the paucity of proof, I understand totally. I was simply trying to fill in some of the ideas; if the reader does not understand some steps, they are free to fill in the details. The details required in a proof are dependent on the intended audience. After all, a proof is nothing more than a means to convince the reader of the truth of a given statement. –  robjohn Aug 22 '11 at 21:29
    
Don't get me wrong, I like your exposition of the argument (and your way of using Jensen is a neat trick that should be explained more often). I agree with you that filling in the details isn't too hard; while doing it here might be the wrong place. Concerning the return key in the comments: Depending on your choice of browser and its additions, you might be able to use this script. I'm pretty happy with it. –  t.b. Aug 22 '11 at 21:42

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