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A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications. (English) Zbl 1282.26025

In 1938, A. Ostrowski obtained a bound for the approximation of a differentiable function by its integral average. If \(f\) is a differentiable real mapping on an interval \(I\), \(a,b\in I\) with \(a<b\) and \(|f'(t)|\leq M\) for all \(t\in[a,b]\), then the inequality \[ \left|f(x)-\frac{1}{b-a}\int_{a}^{b}f(t)dt\right|\leq\left[\frac{1}{4}+ \left(\frac{x-\frac{a+b}{2}}{b-a}\right)^2\right](b-a)M, \] known as the Ostrowski inequality, holds true for every \(x\in[a,b]\). The constant \(\frac{1}{4}\) is the best possible one. In this paper, the authors present some inequalities of Ostrowski type including some weight functions.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
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References:

[1] doi:10.1016/j.aml.2012.02.052 · Zbl 1254.26034 · doi:10.1016/j.aml.2012.02.052
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