How to prove Favard Inequality?

Question

If $f:[a,b]\to \mathbb{R}_{+}$ is a continuous concave function taking non-negative values, and $p>1$ then: \begin{align} \left ( \frac{1}{b-a} \int_a^b f^p(x)dx \right )^{1/p} \leqslant \frac{2}{(p+1)^{1/p}}\left ( \frac{1}{b-a}\int_a^b f(x)dx \right ) \end{align}

Answer 1

Here is a link to the paper from 2003 which could help:

https://www.sciencedirect.com/science/article/pii/S0895717703900483

In particular, see Corollary 4.3, it is exactly your version of Favard's Inequality.