Does this function preserve the order of the nonnegative matrices

Question

Suppose A and B are nonnegative matrices, that is, $A\geq 0$ and $B\geq 0$. Moreover, suppose $\|A\|_{\infty}<1, \|B\|_{\infty}<1$ and $A\leq B$. For $f(x)=x(2-x)$, we know that $f(x)$ is increasing in $(0,1)$. I wonder if it is true that $f(A)\leq f(B)$ if $f(B)\geq 0$.