If $f(x)$ is a continuous function on $\mathbb R$, and $|f(-x)|< |f(x)|$ for all $x>0$. Does it imply that $|f(x)|$ is strictly increasing on $(0,\infty)$?
I tried to use the definition: let $a,b \in (0,\infty)$ with $a, we need to show that $|f(a)|<|f(b)|$. We have $|f(-a)|< |f(a)|$ and $|f(-b)|< |f(b)|$, and I don't know how to proceed!