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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

Citations:

Zbl 0702.26016
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