The Hermite-Hadamard inequality can be easily extended to the case of twice differentiable functions \(f\) with bounded second derivative. Precisely, if \(\gamma\leq f^{\prime\prime} \leq\Gamma,\) then \[ \frac{3S_{2}-2\Gamma}{24}(b-a)^{2}\leq\frac{1}{b-a}\int_{a}^{b}f\,dt-f\left( \frac{a+b}{2}\right) \leq\frac{3S_{2}-2\gamma}{24}(b-a)^{2} \] and \[ \frac{3S_{2}-\gamma}{12}(b-a)^{2}\leq\frac{f(a)+f(b)}{2}-\frac{1}{b-a}\int _{a}^{b}f\,dt\leq\frac{\Gamma}{12}(b-a)^{2} \] The paper under review contains extensions of the Hermite-Hadamard inequality to the context of functions with bounded derivatives of \(n\)th order. For example, if \(f:[a,b]\rightarrow\mathbb{R}\) is an \(n\)-times differentiable function with \(\gamma\leq f^{(n)}\leq\Gamma,\) then it is proved that \[ \frac{(b-a)^{n+1}}{n!2^{n}}\left[ S_{n}+\left( \frac{1+(-1)^{n}} {2(n+1)}-1\right) \Gamma\right]\leq (-1)^{n}\int_{a}^{b}f\,dt \] \[ +\sum_{i=0}^{n-1}\frac{(b-a)^{n-i}}{(n-i)!}\frac{(-1)^{n+1}+(-1)^{i} }{2^{n-i}}f^{(n-i-1)}\biggl(\frac{a+b}{2}\biggr)\leq \frac{(b-a)^{n+1}}{n!2^{n}}\left[ S_{n}+\left( \frac{1+(-1)^{n} }{2(n+1)}-1\right) \gamma\right] \] where \(S_{n}=\frac{f^{(n-1)}(b)-f^{(n-1)}(a)}{b-a}.\) Further extensions are obtained via the concept of harmonic sequence of polynomials.