Ostrowski type inequalities and some selected quadrature formulae

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D. S. Mitrinovic, who left us 25 years ago. His significant inuence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modifications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric in-equality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).

Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshev function and the Grüss inequality are presented.

Refinements of the majorization-type inequalities via green and fink identities and related results

In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral $\int_a^b {f(t)\;du(t)} $, where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

Perturbed Companions of Ostrowski’s Inequality for Absolutely Continuous Functions (I)

Perturbed companions of Ostrowski’s inequality for absolutely continuous functions whose derivatives are either bounded or of bounded variation and applications are given.

New bounds for the companion of Ostrowski’s inequality and applications

In this paper we establish some new bounds for the companion of Ostrowski?s inequality for the case when f??L1[a, b], f???L2[a,b] and f??L2[a, b], respectively. We point out that the results in the first and third cases are sharp and that some of these new estimations can be better than the known results.