If $f,g>0$, f increasing and g decreasing on interval $(0,1)$ then, what conditions imposed for that $fg$ is decreasing in $(0,1)$?

Question

The question is about of the conditions over $f$ and $g$ for the next:

If $f,g>0$, $f$ increasing, $g$ decreasing then what conditions imposed for that $fg$ is decreasing in $(0,1)$?

Where $f,g,fg: (0,1)\rightarrow \mathbb{R}$

Answer 1

If $f$ and $g$ are derivable functions, than we want: $$ (fg)'=f'g+fg'<0 $$

Since $f'g'<0$ this means: $$ \frac{g}{g'}+\frac{f}{f'}>0 \iff \frac{g}{g'}>-\frac{f}{f'} $$

That can be useful if we know something more about the two function.