I am trying to prove the this called "Lyapunov's inequality":
$$\big\||f_0|^{p_0(1-\theta)/p}|f_1|^{p_1\theta/p}\big\|^p_p\leq\|f_0\|^{(1-\theta)p_0}_{p_0}\|f_1\|^{p_1\theta}_{p_1}$$
which is stated on Wikipedia see.
I do not even understand which are $f_0$ and $f_1$ and to which $L^p$ space they belong.
Does anyone have any reference on this particular inequality (preferably with a proof)?
@Mittens has given all the details in his comments, but it seems that a more formal treatment is wanted.
Proposition. Let $(X, {\mathcal S}, \mu)$ be a measure space. Let $p_0, p_1, p\in [1, \infty)$, and $\theta\in (0, 1)$. Let $f_0\in L^{p_0}(X)$ and $f_1\in L^{p_1}(X)$. Then $$ \big\| |f_0|^{\frac{p_0(1-\theta)}{p}}\cdot |f_1|^{\frac{p_1\theta}{p}}\big\|_p^p\leq \big\|f_0\big\|_{p_0}^{p_0(1-\theta)}\big\|f_1\big\|_{p_1}^{p_1\theta}. $$
Proof. Noting that $\|f\|_p^p=\int |f|^p$ and applying the Holder inequality by $(1-\theta)+\theta=1$, we have
\begin{align} \big\| |f_0|^{\frac{p_0(1-\theta)}{p}}\cdot |f_1|^{\frac{p_1\theta}{p}}\big\|_p^p &= \int |f_0|^{p_0(1-\theta)} |f_1|^{p_1\theta}\\ &\leq \Big(\int \big(|f_0|^{p_0(1-\theta)}\big)^{\frac{1}{1-\theta}}\Big)^{1-\theta} \Big(\int \big(|f_1|^{p_1\theta}\big)^{\frac{1}{\theta}}\Big)^{\theta}\\ &=\Big(\int |f_0|^{p_0}\Big)^{1-\theta} \Big(\int |f_1|^{p_1}\big)^{\theta}\\ &=\big\|f_0\big\|_{p_0}^{p_0(1-\theta)} \big\|f_1\big\|_{p_1}^{p_1\theta}. \end{align}
Interpolation. In the Wikepedia page you refer to, only in the next step, we ask $1\leq p_0<p_1<\infty$, $p=(1-\theta)p_0 + \theta p_1$, and $f_0=f_1=f$ to get \begin{equation}\label{finite} \|f\|_p^p\leq \|f\|_{p_0}^{p_0(1-\theta)}\|f\|_{p_1}^{p_1\theta}. \end{equation} Then it is concluded that if $f\in L^{p_0}\cap L^{p_1}$, then $f\in L^p$ for all $p_0<p<p_1$.
A little more. It is easy to show that the interpolation also holds when $p_1=\infty$. That is, if $f\in L^{p_0}\cap L^\infty$, then $f\in L^p$ for all $p_0<p<\infty$.
The proof is simply that by $|f|\leq \|f\|_\infty$ a.e., we have $$ \int |f|^p = \int |f|^{p_0} |f|^{p-p_0} \leq \|f\|_\infty^{p-p_0}\int |f|^{p_0}. $$ In the norm notation, this is $$ \|f\|_p^p\leq \|f\|_{p_0}^{p_0}\|f\|_\infty^{p-p_0}. $$ This matches the above $p_1<\infty$ case, by letting $\theta=0$, and noting $p_1\theta = p - (1-\theta)p_0$ from $p=(1-\theta)p_0 + \theta p_1$.