When is $f(|x|) \leq |f(x)|?$
If $f(|x|) \leq |f(x)|$, what does this say about $f$?
Where $f$ is a real-valued function
I don't have much input on this, but I thought of this from having to show:
$$\sinh (|y|) \leq |\sin z|$$
for $z:= x+iy$ where I had
$$|\sin z| = \sqrt{\sin ^2x + \sinh ^2 y} \geq \sqrt{ \sinh ^2 y} = |\sinh y|$$
which suggests that $|\sinh y| \geq \sinh |y|$ for all $y\in \mathbb{R}$.