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

There is no general result available. You can find examples where the product is increasing, decreasing, and none of the previous. Signs do play a big role here. — b00n heT, Oct 20 '16 at 19:20

score 0 · Answer 1 · answered Oct 20 '16 at 19:35

0

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.

answered Oct 20 '16 at 19:35

Emilio Novati

Maybe the last condition is better if we invert? $(\log g)' < -(\log f)'$? – Ted Shifrin Oct 20 '16 at 20:37
It's more elegant, but I don't know if it's more useful. – Emilio Novati Oct 20 '16 at 20:50
I guess I have problems with zeroes of the functions. But I was thinking that saying that $\log g$ grows slower than $\log(1/f)$ was a pretty intuitive thing. No big deal. :) – Ted Shifrin Oct 20 '16 at 20:57

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