On the Ostrowski's integral inequality for mappings with bounded variation and applications (Q2711480)

The author proves the following Ostrowski's type inequality. Let \( u:[a,b]\rightarrow \Re \) be a mapping with bounded variation on \([a,b]\) . Then for all \(x\in [a,b]\) we have the inequality NEWLINE\[NEWLINE \left|\int_{a}^{b}u(t)dt-u(x)(b-a)\right|\leq \left( \frac{b-a}{2}+\left|x-\frac{a+b}{2}\right|\right) \bigvee_{a}^{b}(u), NEWLINE\]NEWLINE where \(\bigvee\limits_{a}^{b}(u)\) denotes the total variation of \(u\) . Applications in obtaining a Riemann's type quadrature formula and for Euler's Beta function are also given.