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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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One can look at log-concave distributions. In particular, this article mentions an application of near log-concave distributions to learning.

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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… is devoted to them. (See the link for definition.) – Martin Sleziak Mar 30 '11 at 16:38

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