For $x\_+ \in (0,\infty)$ let $f\colon(0,x\_+] \to (0,\infty)$ be a continous differentiable function with $f(x) > 0$ and $f'(x) < 0$ for all $x \in (0,x\_+]$.
Moreover, we assume that
$$\lim\_{x \to 0} f(x) = \infty$$
holds.


**The question:** Does this implies that there exists a $\beta \in (0,\infty)$ such that $f(x)f(y) \ge \beta f(xy)$ for all $x,y \in (0,x\_+]$.


From my intuition this is not valid, however i am not able to derive a counter example; that function needs to be really steep in direction $x \to 0$.



Positive, monotone decreasing function, with limit in 0 equal to ∞ submultiplicative up to an factor?
A counterexample is given by
$$f(x)=e^{1/x}.$$


Indeed, then $f(x)f(x)/f(xx)\to0$ as $x\downarrow0$, so that, for any real $x\_+$, there is no real $\beta>0$ such that $f(x)f(y)\ge\beta f(xy)$ for all $x,y$ in $(0,x\_+]$


