We know that for $b \geq a$, and some $s \geq 0$, a concave function $f$ satisfies:
$f(a+s) - f(a) \geq f(b+s) - f(b)$.
This is not a frequent definition of concavity, but can be found, e.g., here.
My question is, what is the name of this property? Is it easy to prove without Taylor Expansions?
The property you are referring to simply states that the directional derivative of a 1-dimensional concave function is non-increasing. You should be able to find a proof for that inequality here. The link proves it for convex functions, for concave functions simply reverse the sign of the inequality. I was unable to find a standard name for the inequality though, and yes it is easy to prove.
