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Some perturbed inequalities of Ostrowski type for high-order differentiable functions and applications

Erden Samet, Sarikaya Mehmet Zeki

Journal of Applied Analysis

2021

Vol. 27, nr 1

57--64

We firstly establish inequalities for functionswhose high degree derivatives are convex via an equality which was presented previously. Then we derive inequalities for functions whose high-order derivatives are absolutely continuous by using the same equality. In addition,we examine connections between inequalities obtained in earlierworks and our results. Finally, some estimates of composite quadrature rules are given.

The Nemitskij operator on Lipk-type and BVk-type spaces

Giménez J., López L., Merentes N.

Demonstratio Mathematica

2013

Vol. 46, nr 3

543--558

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of R. We obtain a result concerning the integrability of products of the form (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.

Integral inequalities associated with inequalities of Opial type

Florkiewicz B., Kuchta M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 1

41-45

We derive some new integral inequalities of the form: [...], h [belongs to] H, where I = (alpha, beta), -[infinity is less than or equal alpha < beta is less than or equal infinity], p > 0, H is a wide class of absolutely continuous functions h defined on I and satisfying the limit conditions h(alpha) = 0 or h(beta) = 0, the functions r, s and u are any set of functions related by the appropriate weight functions. To get the desired inequality, at first we derive an integral inequality of Opial type using a uniform method of obtaining integral inequalities with weight functions involving the function and its derivative [2].