Consider $g$ such that it is
- continuous, strictly increasing, and convex
and two points $0<a<b$
If $h_{a,b}(C)$ is the function that scales these two points two points, applies $g$, and then takes the ratio of these values, i.e.
$$
h_{a,b}(C) = \frac{g(C*a)}{g(C*b)}
$$
can we prove whether $h$ will be convex, concave, or neither? How about whether it is increasing/decreasing?
Or will the result depend on $g$? (looking at This answer, it seems that perhaps it can go either way? But here we have a ratio of a function to itself, just at a different point. Maybe this information allows us to say more)?
A comment: I have tried taking the derivative of $h$, i have not been able to get anywhere with it.
Another comment: I feel pretty strongly that $h$ should have to be decreasing (and also that it should be convex, but less strongly so about that).
My reason is as follows: Consider two linear functions, with slopes $m_1<m_2$. Then the ratio between any two points $a<b$, with $a$ on the flatter line, is $\frac{m_1 a}{m_2 b}$. If we increasing $m_2$ then this decreasing.
- Wouldn't scaling two points along a convex function be similar to scaling two points along these lines, except that the slopes of the lines are changing as well. However, by convexity, the slope of the "steeper" line is changing faster than the slope of the flatter line.
Edit: Restrict $C$ to be in $(0,\infty)$.
(partly because I forgot to include this, and partly because an answer has been provided for when $C$ can be negative)
Best Answer
$h$ does not need to be convex, concave, increasing or decreasing. An example is $g(C) = 1 + e^C$, $a=1$ and $b=2$.