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On Generalization of Dragomir’s Inequalities

Hüseyin Budak*, Mehmet Zeki Sarıkaya

Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

Abstract: In this paper, we establish some generalization of weighted Ostrowski type integral inequalities for functions of bounded variation.

Keywords: Function of bounded variation, Ostrowski type inequalities, Riemann-Stieltjes integrals

Received August 15, 2017; Revised September 16, 2017; Accepted September 21, 2017

Citation: Hüseyin Budak, Mehmet Zeki Sarıkaya, On Generalization of Dragomir’s Inequalities. Turkish Journal of Analysis and Number Theory, 5 (5). 191-196. January 2017. doi:10.12691/tjant-5-5-5

1. Introduction

Let be a differentiable mapping on whose derivative is bounded on i.e. Then we have the inequality

[picture] | (1)

for all [16]. The constant is the best possible. This inequality is well known in the literature as the Ostrowski inequality.

Definition 1. Let be any partition of and let Then is said to be of bounded variation if the sum

is bounded for all such partitions.

Let be of bounded variation on , and denotes the sum corresponding to the partition of . The number

is called the total variation of on Here denotes the family of partitions of

In [7], Dragomir proved following Ostrowski type inequalities related functions of bounded variation:

Theorem 1. Let be a mapping of bounded variation on Then

holds for all The constant is the best possible.

In [9], Dragomir gave a simple proof of following Lemma:

Lemma 1. Let If is continious on and is bounded variation on then

In [5], Dragomir obtained following Ostrowski type inequality for functions of bounded variation:

Theorem 2. Let be a division of the interval and be points so that If is of bounded variation on then we have the inequality:

[picture] | (2)

where and is the total variation of on the interval

For some recent results connected with functions of bounded variation see [1,2,3,4,6,8,10,11,12,13,14,15,17,18,19,20,21].

The aim of this paper is to obtain some generalization of weighted Ostrowski type integral inequalities for functions of bounded variation.

2. Main Results

Firstly, we will give the following notations which are used in main Theorem:

Let be a partition of the interval , be points so that Let be continious and positive mapping on , and

Theorem 3. If is of bounded variation on then we have the inequalities

[picture] | (3)

and

[picture] | (4)

where is the total variation of on the interval

Proof. Let us consider the functions defined by

Integrating by parts , we obtain

[picture] | (5)

In last equality in (5), we have

[picture] | (6)

and similarly

[picture] | (7)

Adding (6) and (7) in (5), we get the equality

[picture] | (8)

On the other hand, taking modulus in (8) and using triangle inequality we have

[picture] | (9)

Using Lemma 1 in last inequality in (9), we have

[picture] | (10)

Putting (10) in (9), we obtain

[picture] | (11)

This completes the proof of first inequality in (3).

On the other hand, in last inequality in (11), we have

[picture] | (12)

Adding (12) in last inequality in (11), we obtain the inequality (3).

Finally, for proof of inequality (4), taking modulus in (8), we have

[picture] | (13)

Using Lemma 1 for the last integral of (13), we have

[picture] | (14)

Adding (14) in (13), we obtain

[picture] | (15)

which completes the proof of first inequality in (4).

Using triangle inequality in last inequality in (15), we have

and

This completes the proof.

Remark 1. Under assumptions Theorem 3 with , the inequality (3) reduces inequality (2).

Remark 2. If (differantiable with respect to ) in Theorem 3, then we have the inequality

[picture] | (16)

which was proved by Kuei-Lin Tseng et al. in [20].

Remark 3. If we choose in (16), inequality reduces inequality (2).

Corollary 1. Under assumption Theorem 3, choosing in inequality (4) we obtain the inequality

[picture] | (17)

Remark 4.

1) In (17), if we take then we have the "weighted left rectangle inequality"

2) If we take in (17) then we have the "weighted right rectangle inequality"

3. Applications for Quadrature Rule

Let us consider the arbitrary division

and let be continious function with

Then the following Theorem holds.

Theorem 4. Let is of bounded variatin on and Then we have the quadrature formula:

The remainder term satisfies

[picture] | (18)

Proof. Applying Corollary 1 to interval we have the inequality

[picture] | (19)

Summing the inequality (19) over from to , then we have

This completes proof of the first inequality in (18).

Also, we have

and

which completes the proof.

Remark 5.

1) If we choose then we have the weighted left rectangle rule

The remainder satisfies

2) Similarly, choosing we have the weighted right rectangle rule

And, the remainder term satisfies

References

[1]   | Alomari, M. W., “A Generalization of weighted companion of Ostrowski integral inequality for mappings of bounded variation,” RGMIA Research Report Collection, 14, 2011, Article 87, 11 pp.
[2]   | Alomari, M. W. and Latif, M.A. “Weighted companion for the Ostrowski and the generalized trapezoid inequalities for mappings of bounded variation,” RGMIA Research Report Collection, 14, 2011, Article 92, 10 pp.
[3]   | Cerone, P. Cheung, W.S. and Dragomir, S.S., “On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation,” Computers and Mathematics with Applications, 54, 2007, 183-191.
[4]   | P. Cerone, S. S. Dragomir, and C. E. M. Pearce, A generalized trapezoid inequality for functions of bounded variation, Turk J Math, 24 (2000), 147-163.
[5]   | Dragomir, S. S., “The Ostrowski integral inequality for mappings of bounded variation,” Bull.Austral. Math. Soc., 60(1), 1999, 495-508.
[6]   | Dragomir, S. S., “On the midpoint quadrature formula for mappings with bounded variation and applications,” Kragujevac J. Math. 22, 2000, 13-19.
[7]   | Dragomir, S. S., “On the Ostrowski's integral inequality for mappings with bounded variation and applications,” Math. Inequal. Appl. 4(1), 2001, 59-66.
[8]   | Dragomir, S. S., “A companion of Ostrowski's inequality for functions of bounded variation and applications,” Int. J. Nonlinear Anal. Appl. 5(1), 2014, 89-97.
[9]   | Dragomir, S. S., “Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation.” Arch. Math. (Basel) 91(5), 2008, 450-460.
[10]   | Dragomir, S.S. and Momoniat, E., “A Three point quadrature rule for functions of bounded variation and applications,” RGMIA Research Report Collection, 14, 2011, Article 33, 16 pp.
[11]   | Dragomir, S. S., “Some perturbed Ostrowski type inequalities for functions of bounded variation,” RGMIA Res. Rep. Coll. 16, 2013, Art. 93.
[12]   | Liu, W. and Sun, Y., “A Refinement of the companion of Ostrowski inequality for functions of bounded variation and Applications,” arXiv:1207.3861v1, 2012.
[13]   | Mishra, V.N., “Some problems on approximations of functions in banach spaces,” Ph.D. Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.
[14]   | Mishra V.N. and Mishra, L.N., “Trigonometric approximation of signals (Functions) in LP (p≥1)-norm,” International Journal of Contemporary Mathematical Sciences, 7(19), 2012, 909-918.
[15]   | Mishra, L.N., “On existence and behavior of solutions to some nonlinear integral equations with Applications,” Ph.D. Thesis (2017), National Institute of Technology, Silchar 788010, Assam, India.
[16]   | Ostrowski, A., M. “Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert,” Comment. Math. Helv. 10, 1938, 226-227.
[17]   | Tseng, K-L, Yang, G-S and Dragomir, S. S., “Generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications,” Mathematical and Computer Modelling, 40, 2004, 77-84.
[18]   | Tseng, K-L, “Improvements of some inequalites of Ostrowski type and their applications,” Taiwan. J. Math. 12(9), 2008, 2427-2441.
[19]   | Tseng, K-L, Hwang, S-R, Yang, G-S and Chou, Y-M, “Improvements of the Ostrowski integral inequality for mappings of bounded variation I,” Applied Mathematics and Computation 217, 2010, 2348-2355.
[20]   | Tseng, K-L, Hwang, S-R, Yang, G-S and Chou, Y-M, “Weighted Ostrowski ,ntegral ,nequality for mappings of bounded variation,“ Taiwanese J. of Math., 15(2), 2011, 573-585.
[21]   | Tseng, K-L, “Improvements of the Ostrowski integral inequality for mappings of bounded variation II,” Applied Mathematics and Computation 218, 2012, 5841-5847.