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Inequalities of Hadamard’s type for Lipschitzian mappings and their applications. (English) Zbl 0956.26015

The authors prove the following two inequalities of Hadamard’s type: Let \(f:I\subseteq \mathbb{R\rightarrow R}\) be an \(M\)-Lipschitzian mapping on \(I\) and \(a,b\in I\) with \(a < b\). Then we have the inequalities \[ \left|f\left( \frac{a+b}{2}\right) -\frac{1}{b-a}\int_{a}^{b}f(x) dx\right|\leq \frac{M}{4}(b-a) \] and \[ \left|\frac{f(a)+f(b)}{2}- \frac{1} {b-a} \int_{a}^{b}f(x) dx\right|\leq \frac{M}{3}(b-a). \] Some applications for means of two positive numbers are also given.

MSC:

26D15 Inequalities for sums, series and integrals
26E60 Means
26A16 Lipschitz (Hölder) classes
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