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

Alzer, Horst

C. R. Math. Acad. Sci., Soc. R. Can. 11, No. 6, 255-258 (1989).

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

Keywords:

Jensen-Hadamard integral inequalities; convex functions; exponential function

Citations:

Zbl 0702.26016