I have come across a needed condition on a continuous probability distribution defined over $[0, \infty]$ and wonder whether it has a name. For a distribution with CDF $F$ and pdf $f$ I need that the quantity: $$\phi(x) \equiv \frac{f(x)}{F(x)+xf(x)}$$ be monotonically non-increasing. Putting the condition in another form (by taking derivatives), the requirement is that for all $x \in [0,\infty]$ such that $f'(x) > 0$: $$f(x) \geq \sqrt{\frac{F(x) f'(x)}{2}}$$

This seems to be a commonly satisfied property, so does it have a name? It is related to, but distinct from a monotone hazard rate condition.