  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 $j$th
derivative of $f$. In this note we prove Hermite-Hadamard inequalities for
$(p,a,b)$-convex functions that are significantly tighter than the classical
Hermite-Hadamard inequality. We also prove inequalities for fractional
integrals that involve $(p,a,b)$-convex functions.
