Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation (Q999101)

The author gives some refinements of the inequalities \[ \left|\int_a^bf(t)dt-f(x)(b-a)\right|\leq\left(\frac{b-a}{2}+\left|x -\frac{a+b}{2}\right|\right)\bigvee_a^b(f), \] and \[ \left|\int_a^bf(t)dt-\Big(f(a)(x-a)+f(b)(b-x)\Big)\right| \leq\left(\frac{b-a}{2}+\left|x-\frac{a+b}{2}\right|\right)\bigvee_a^b(f), \] where \(f\) is a function of bounded variation on \([a,b]\), and \(\bigvee_a^b(f)\) denotes the total variation of \(f\) on \([a,b]\). The author, also gives a refinement of Ostrowski's inequality for functions of bounded variation.