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