As you know, Hermite-Hadamard Integral inequalities gives lower and upper bounds for integrals of convex functions.
Now, let's assume that $f(x)$ and $f^{k}(x)$($k$ th derivate of $f(x)$) are convex functions on the interval $[a,b]$. Is there a refinement of Hermite- Hadamard inequalities for
$$\int^b_a f(x) dx?$$
There are many improvements, enhancements and analogies of this celebrated inequality. The exhausting study is given by Dragomir, Peirce and Pecaric in this monograph http://rgmia.org/papers/monographs/Master.pdf
Also Croatian Pecarić's team deals with inequalities of HH type. Many papers are published in Mathematical Inequalities and Applications.
If you have an access, you could consult Mathematical Reviews (https://mathscinet.ams.org/mathscinet/).
During my reviewing career I have seen many inequalities of this type. But there are too many of them to give the precise answer. Here I would like to say that such results almost surely exist.