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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 $y\in \mathbb{R}$.
split the cases:
Nonnegative $x$:
For nonnegative values of $x$, you have $f(|x|)=f(x)$ which means that $f(|x|)\leq|f(x)|$ if and only if $f(x)\leq |f(x)|$ which is always true.
negative $x$:
For negative values of $x$, the inequality $f(|x|)\leq |f(x)|$ translates to $$f(-x) \leq |f(x)|$$ which, for example, is true for odd functions like $\sinh$ since if $f(-x)=-f(x)$, then $f(-x)=-f(x)\leq |f(x)|$.
