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.
$\int f \log f d\mu \le 2 [\int|f-1|^p d\mu]^{1/p}+\frac{2}{p-1}\int |f-1|^p d\mu, p>1$ $$ \int f \log f d\mu \le 2 \left[\int|f-1|^p d\mu\right]^{1/p}+\frac{2}{p-1}\int |f-1|^p d\mu,\quad p>1 $$ where f$f$ is a probability density.
