I have a sequence of integers meeting the following inequality:
$u_n \leq \frac{u_{n-1}+u_{n+1}}{2} + \frac{1}{2}$. In other words, the sequence is "approximately convex", and the difference comes essentially from the fact that we are working with integers.
For example, (2, 2, 1, 1) is not convex but is "approximately convex" in the above sense.
It looks like $\frac{1}{2}$-Jensen-convexity, but I am not sure.