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A function $f:[a,b] \rightarrow \mathbb{R}$ is called $(p,a,b)$-convex if $f$is $p$ times continuously differentiable, $f^{(p)}$ is convex and increasing,and $f^{(k)}(a)=0$ for all $k=1,\ldots,p$ where $f^{(j)}$ is the $j$thderivative of $f$. In this note we prove Hermite-Hadamard inequalities for$(p,a,b)$-convex functions that are significantly tighter than the classicalHermite-Hadamard inequality. We also prove inequalities for fractionalintegrals that involve $(p,a,b)$-convex functions.

Hermite-Hadamard inequalities for (p,a,b)-convex functions