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American Journal of Public Health Research
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The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
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The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
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Clinical and Histological Features in Wegener Granulomatosis
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The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

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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

Institute of Southern Punjab, Multan-Pakistan

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.

Keywords: Ostrowski's inequality, functions of bounded variations, numerical integration

Received November 24, 2021; Revised December 29, 2021; Accepted January 09, 2022

Citation: M. A. Mustafa, A. Qayyum, T. Hussain, M. Saleem, Some Integral Inequalities for the Quadratic Functions of Bounded Variations and Application. Turkish Journal of Analysis and Number Theory, 10 (1). 1-3. January 2022. doi:10.12691/tjant-10-1-1

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

[picture] | (2.1)

Where

and represents the total variation of on

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

[picture] | (2.2)

Integrating by parts, we get the following identity

By using (2.2), we have

[picture] | (2.3)

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

[picture] | (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

[1]   | A. Ostrowski, Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10, No. 1, pp. 226-227, 1938.
[2]   | A. Qayyum, M. Shoaib and S. Erden, On generaliźed fractional Ostrowski type inequalities for higher order derivatives, Communication in Mathematical Modeling And Applications, Vol. 4 (2), 2019.
[3]   | A. Qayyum, M. Shoaib and I. Faye. On New Weighted Ostrowski Type inequalities Involving Integral Means over End Intervals and Application, Turkish Journal of Analysis and Number Theory, 3(2): 61-67, 2015.
[4]   | A. R. Kashif, T. S. Khan, A. Qayyum and I. Faye, A comparison and error analysis of error bounds, International Journal of Analysis and Applications, 16 (5), 2018.
[5]   | M. Iftikhar, A. Qayyum, S. Fahad and M. Arslan, A new version of Ostrowski type integral inequalities for different differentiable mapping, Open J. Math. Sci. Vol. 5(1), pp. 353-359, 2021.
[6]   | S. Obiedat, M. A. Latif and A. Qayyum, Ostrowski type inequality using a 5-step weighted kernel, Internatioal Journal of Analysis and Applications, 13(3), 2019.
[7]   | S. Obiedat, M. A. Latif and A. Qayyum, A weighted companion of Ostrowski’s inequalty using three step weighted kernel, Miskolc Mathematical Notes, Vol. 20, 2019.
[8]   | S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bulletin of the Australian Mathematical Society, 60(1), pp. 495-508, 1999.
[9]   | S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications, Kragujevac J. Math. 22, pp. 13-19, 2000.
[10]   | S. S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Mathematical Inequalities & Applications, 4, No.1, pp. 59-66, 2001.
[11]   | S. S. Dragomir, Refinements of the generalised trapeźoid and Ostrowski inequalities for functions of bounded variation, Arch. Math. (Basel) 91, No. 5, pp. 450-460, 2008.
[12]   | S. S. Dragomir, A companion of Ostrowski’s inequality for functions of bounded variation and applications, International Journal of Nonlinear Analysis and Applications, 5, No. 1, pp. 89-97, 2014.
[13]   | P. Cerone, S. S. Dragomir and C. E. M. Pearce, A generaliźed trapeźoid inequality for functions of bounded variation, Turkish J. Math. 24, No. 2, pp. 147-163, 2000.
[14]   | W. Liu and Y. Sun, A refinement of the companion of Ostrowski inequality for functions of bounded variation and Applications, arXiv:1207.3861v1, 2012.
[15]   | H. Budak, M. Z. Sarikaya and A. Qayyum, Improvement in companion of Ostrowski type inequalities for mappings whose first derivatives are of bounded variation and applications, Filomat, 31: 13, 2017.
[16]   | H. Budak and M. Z. Sarikaya and A. Qayyum, New refinements and applications of Ostrowski type inequalities for mappings whose nth derivatives are of bounded variation, TWMS J. App. Eng. Math. V.11, N.2, pp. 424-435, 2021.
[17]   | H. Budak and M. Z. Sarikaya, A new Ostrowski type inequality for functions whose first derivatives are of bounded variation, Moroccan Journal of Pure and Applied Analysis, Vol. 2(1), pp. 1-11, 2016.
[18]   | H. Budak and M. Z. Sarikaya, A companion of Ostrowski type inequalities for mappings of bounded variation and some applications, Transactions of A. Raźmadźe Mathematical Institute, 171(2), pp. 136-143, 2017.
[19]   | H. Budak and M. Z. Sarikaya, New generaliźed inequalities for functions of bounded variation, Cumhuriyet Sci. J., Vol. 39-3, pp. 668-678, 2018.