## A note on Hadamard’s inequalities.(English)Zbl 0707.26012

Related to a result of the reviewer [Elem. Math. 43, 177-180 (1988; Zbl 0702.26016)], the author proves that if $$f:[a,b]\to {\mathbb{R}}$$ is a 2k-times differentiable function $$(k\geq 1)$$ with $$f^{(2k)}(x)\geq 0$$ for $$x\in (a,b),$$ then $\frac{1}{b-a}\int^{b}_{a}f(x)dx\leq \frac{1}{2}\cdot \sum^{2k-2}_{i=0}\frac{(b-a)^ i}{(i+1)!}[f^{(i)}(a)+(-1)^ if^{(i)}(b)].$ He then obtains an interesting application to the exponential function.
Reviewer: J.Sándor

### MSC:

 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations

Zbl 0702.26016