# Properties of nearly concave functions

Is there any existing literature on the properties/applications of the following class of functions?

$$\frac{f(E[x])}{E[f(x)]}\geq c$$

where $c< 1$ is a constant. Note that for $c=1$ these are exactly concave functions.

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$\beta$-log concave functions in that article seems to be related to related to $\varepsilon$-convex function in the sense that they are logarithms of them. Chapter "Approximately Convex Functions" of the book Stability of functional equations in several variables By Donald H. Hyers, George Isac, Themistocles M. Rassias books.google.com/… is devoted to them. (See the link for definition.) –  Martin Sleziak Mar 30 '11 at 16:38