Concave down increasing example I'm looking for a concave down increasing-function, see the image in the right lower corner.
Basically I need a function f(x) which will rise slower as x is increasing. The x will be in range of [0.10 .. 10], so f(2x) < 2*f(x) is true. Also if
I would also like to have some constants which can change the way/speed the function is concaving.
Any suggestion on a formula?


If you are restricted to a positive $x$, you could use
$$ f(x) = a \sqrt{x}, \quad a > 0 $$
which satisfies
$$ f(2 \cdot x) = a \sqrt{2 \cdot x} = \sqrt{2} \cdot a \sqrt{x} = \sqrt{2} \cdot f(x) < 2 \cdot f(x) $$
