Can a convex/concave function be transformed to a concave/convex function in a generic way? My question is that let’s say I have a concave increasing function, can I transform it into a new convex increasing function? I know you can take log or exp of the function till it changes one to another, but I mean in a more generic and certain way.
 A: Here is one idea, surely not what you intended. 
Negation flips about the horizontal, changing
concave to convex.
Reflecting in a vertical line $x=c$ then turns convex decreasing to
convex increasing.
Shifting vertically with constants retains these properties.

          


          

$3+x^2$ is concave increasing, $10-x^2$ is convex decreasing.
$10-(x-4)^2$ reflects around the vertical $x=4$.


A: If we assume that $f$ is strictly increasing and concave it follows that it has an inverse $f^{-1}$ that is strictly increasing.

*

*$f^{-1}$ is convex because for all $x,y$ in the range of $f$ we have $u,v$ such that $x=f(u),y=f(v)$ and
$$
f^{-1}\Big(\frac{x+y}{2}\Big)=f^{-1}\Big(\frac{f(u)+f(v)}{2}\Big)\,.
$$
Note that
$$
f^{-1}\Big(\frac{x+y}{2}\Big)\le \frac{f^{-1}(x)+f^{-1}(y)}{2}\,,
$$
which is the convexity of $f^{-1}\,,$
is equivalent to
$$
\frac{x+y}{2}\le f\Big(\frac{f^{-1}(x)+f^{-1}(y)}{2}\Big)\,,
$$
that is, to
$$
\frac{f(u)+f(v)}{2}\le f\Big(\frac{u+v}{2}\Big)\,,
$$
which is the concavity of $f$.

