Hermite-Hadamard inequality for convex stochastic processes (Q662371)

The author extends the Hermite-Hadamard inequality to convex stochastic processes. In particular, it was proved that if the stochastic process \(X:I\times \Omega \rightarrow \mathbb{R}\) is Jensen-convex and mean square continuous in the interval \(I\), then \[ X\left( \frac{u+v}{2},.\right) \leq \frac{1}{v-u}\int _{u}^{v}X(t,.)dt\leq \frac{X(u,.)+X(v,.)}{2}\text{ \;\;a.e.}, \] for all \(u,v\in I\). Furthermore, the converse of the above result is also derived and proved.