Mathworld defines a convex function as
(...) a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.
Is there any name for a function for which the above is true, but only for intervals larger than sone value. Maybe something like that:
A < insert-function-name-here > is a continuous function $f$ for which there exists such a (finite) value of $p\in\mathbb{R^+}$ such that the value at the midpoint of every interval larger than $p$ in the domain of $f$ does not exceed the arithmetic mean of its values at the ends of the interval.
Specifically I'm thinking about functions such as $$f(x) = x/20 - x^2/80 + x^4/20000$$ in the context of optimisation. Basically functions which go to infinity towards the edges of their domain and have a general 'dip' in the middle. The above function is not convex, but it can be optimised and its (global) minimum can be easily found. $-f(x)$ is also not convex and it cannot be optimised.
Maybe something like 'wide-sense convex'? I'd prefer to use an established term rather than come up with my own though.