Hermite–Hadamard inequality for semiconvex functions of rate $(k_1,k_2)$ on the coordinates and optimal mass transportation

Abstract

We give a new Hermite–Hadamard inequality for a function $f:\left[a,b\right]\times \left[c,d\right]\subset {ℝ}^2\to ℝ$ which is semiconvex of rate $\left(k_1,k_2\right)$ on the coordinates. This generalizes some existing results on Hermite–Hadamard inequalities of S. S. Dragomir. In addition, we explain the Hermite–Hadamard inequality from the point of view of optimal mass transportation with cost function $c\left(x,y\right):=f\left(y-x\right)+\frac{k_1}{2}|x_1-y_1{|}^2+\frac{k_2}{2}|x_2-y_2{|}^2$, where $f\left(\cdot \right):\left[a,b\right]\times \left[c,d\right]\to \left[0,\infty \right)$ is semiconvex of rate $\left(k_1,k_2\right)$ on the coordinates and $x=\left(x_1,x_2\right)$, $y=\left(y_1,y_2\right)\in \left[a,b\right]\times \left[c,d\right]$.