﻿ Some Integral Inequalities for the Quadratic Functions of Bounded Variations and Application
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Some Integral Inequalities for the Quadratic Functions of Bounded Variations and Application

M. A. Mustafa, A. Qayyum , T. Hussain, M. Saleem
Turkish Journal of Analysis and Number Theory. 2022, 10(1), 1-3. DOI: 10.12691/tjant-10-1-1
Received November 24, 2021; Revised December 29, 2021; Accepted January 09, 2022

Abstract

In this paper, some essential inequalities are established for the quadratic function of bounded variation by using 7-step kernel. Some previous results are recaptured. Applications for quadrature rule and probability density function are also provided.

1. Introduction

In 1938, a Ukrainian Mathematician A. M. Ostrowski derived an inequality 1 which is known as Ostrowski's inequality. After that several mathematicians worked on refinements to increase people’s interest in Ostrowski's inequality 2, 3, 4, 5, 6, 7.

Dragomiŕ et. al. 8, 9, 10, 11, 12 expanded the Ostrowski’s inequality to the broader field of mappings of bounded variations. Several authors have recently addressed the generalization of the Ostrowski's type inequalities for function of bounded variation 13, 14. In this paper, we'll further expand the work of H. Budąk et. al. 15, 16, 17, 18, 19 for the function of bounded variation by using 7-step kernel.

2. Main Results

Theorem 1. Let be such that is a continuous function of bounded variation on Then we have

 (2.1)

Where

and represents the total variation of on

Proof: To prove our required result, first of all we introduce a mapping

 (2.2)

Integrating by parts, we get the following identity

By using (2.2), we have

 (2.3)

It is a fact that if is continuous on and is function of bounded variation on then exists and

 (2.4)

By using (2.4) for each term in (2.3), we get

Hence proved.

Remark 1 By substituting in (2.1), we get

Which was given by H. Budak in 17.

Corollary 1 Under assumption of Theorem 1 with we obtain

Corollary 2 Under assumption of Theorem 1 with we obtain

Corollary 3 Let then by using Theorem 1, we obtain

where

Corollary 4 Let be a Lipschitźian mapping for positive constant then

Proof. Since is Lipschitźiąn on the interval If represents family of partitions on then

Hence proved.

References

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