Product of an increasing and a decreasing function I am interested on finding conditions on a function $f(x)$, $x>0$, such that
$$g(x)=\dfrac{f(x)}{x^3},$$
is a decreasing function. Unfortunately, the function $f$ in my context is not in closed form but I know that it is increasing, positive, and asymptotically bounded. I would like to provide some general, but interpretable, conditions such that $g$ is a decreasing function. For instance, a general condition would be
$$g'(x)<0 \Rightarrow x\dfrac{d}{dx}\log f(x)<3,$$
which, in my opinion, is not very informative. I wonder if there is a better criterion which provides milder or more interpretable conditions on $f$.

If you use either the Quotient Rule in your definition of $ \ g(x) \ $, or carry out the differentiation implied in your second expression, you will arrive at
$$ x \ f'(x) \ < \ 3 \ f(x) \ , $$
since $ \ x^6 > 0 \ $ (I'm assuming this is a real-valued function).  According to your stated conditions, all of the factors are positive, so this gives something of a useful relation between $ \ f(x) \ $ and $ \ f'(x) \ $ . 
