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Generalizations and refinements of Hermite-Hadamard’s inequality. (English) Zbl 1096.26014
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 $f:[a,b]\rightarrow\Bbb{R}$ 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.

MSC:
26D15Inequalities for sums, series and integrals of real functions
41A55Approximate quadratures
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References:
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