I am looking for a weight function $\phi$
- concave
- strict increasing
- differentiable
that solves the following:
$$\phi(25)-\phi(18.25) < \phi(18.25) - \phi(16) \tag{1}$$
The idea is to give greater weight to lower numbers - enough so that (1) is solved.
To my surprise a number of root and logarithm functions didn't do the trick of fulfilling (1). So what other similar weight functions are there?