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.
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 $x=4$.