# An “elementary” inequality

The following is left unproven in a monograph. The author refers to it as "elementary exercise" but I am unable to prove it. Any insight is appreciated. $$\int f \log f d\mu \le 2 \left[\int|f-1|^p d\mu\right]^{1/p}+\frac{2}{p-1}\int |f-1|^p d\mu,\quad p>1$$ where $$f$$ is a probability density.

• You really mean $f$ a probablity density? not $\mu$? Also please quote the monograph. – YCor Sep 5 at 7:03
• @YCor I rather expect that $\mu$ is probability measure. Because the inequality is not homogeneous with respect to $\mu$. – Fedor Petrov Sep 5 at 9:04
• This follows from $(1+x) \log (1+x)\le 2x +2/(p-1)x^p$ for $x \ge 0$ writing $0 \le f=1+g$ and integrating only where $g \ge 0$. – Giorgio Metafune Sep 5 at 10:38
• $\mu$ is indeed a probability measure in the context of the monograph. f is probability density with respect to $\mu$. But seems that it is true in general by Metafune's proof? The inequality is from (13.6) in "Dynamic to random matrix" by Erdos and Yau. – Daniel Li Sep 5 at 17:20
• I used that $\mu$ is a probability measure to have $\|g\|_1 \le \|g\|_p$ and $f \ge 0$. – Giorgio Metafune Sep 5 at 17:37