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Some Ostrowski and trapezoid type inequalities for the Stieltjes integral in the case of Lischitzian integrators for both Hölder continuous and monotoonic integrals are obtained. The dual case is also analysed. Applications for the midpoint rule are pointed out as well.

A companion for the Ostrowski and the generalised trapezoid in-equalites for various classes of functions, including functions of bounded variation , Lipschitzian, convex and absolutely continuous functions is established. Applications for weighted means are also given.

Utilising the Beesack version of the Darst-Pollard inequality, some error bounds for approximating the Riemann-Stieltjes integral are given. Some applications related to the trapezoid and mid-point quadrature rules are provided .

Inequalities of the majorisation type for convex functions and Stieltjes integrals are given. Applications for some particular convex functions of interest are also pointed out.

Some new results that provide refinements and reverses of the Cauchy-Bunyakovsky-Schwarz (CBS) −inequality in the general setting of Measure Theory and under some boundedness conditions for the functions involved are given.

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