True or not $\int_{-\infty}^{\infty}\lvert\log\sqrt{f(x)}\rvert dx\le\frac12\int\limits_{-\infty}^{\infty}\lvert\log\left(f(x)\right)\rvert dx$

Question

Is the following true ?

$$\int\limits\lvert\log\sqrt{f(x)}\rvert dx\le\frac12\int\limits\lvert\log\left(f(x)\right)\rvert dx$$ when $f\ge0$ and is bounded ?

Is this a consequence of a Jensen-type inequality ?

Answer 1

It's an equality: $\log x^{1/2}=\frac12\,\log x $. for $x>0$