Convexity using a combination of derivatives (Q1766452)

The authors deal with the following three generalizations of the second derivative: the forward lower second derivative \[ \underline f^+_2(x)= \liminf_{h\to 0+}(f(x + 2h) - 2f(x + h) + f(x))/h^2; \] the backward lower second derivative \[ \underline f^-_2 = \liminf_{h\to 0+} (f(x) - 2f(x - h) + f(x - 2h))/h^2; \] and the symmetric lower second derivative \[ \underline f^0_2 = \liminf_{h\to 0+} (f(x + h) - 2f(x) + f(x - h))/h^2. \] In the article the relationship of the above derivates to convexity is investigated. It is shown, i.e.: Let \(I\) be an open interval and \(f\) a real measurable function defined on \(I\) such that forward, backward, and symmetric lower derivates are all non-negative on \(I\). Then \(f\) is convex on \(I\).