On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential

We consider the class of functions $u\in C^2((0,\mathrm{\infty }))$ satisfying second-order differential inequalities in the form $u^{″}(x)+\frac{k}{x^2}u(x)\ge 0$ for all $x>0$. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ( $k=\frac{1}{4}$ and $0<k<\frac{1}{4}$). We next extend our obtained results to the two-dimensional case. In the limit case $k\to 0^{+}$ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.