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.