- Research
- Open access
- Published:
Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications
Journal of Inequalities and Applications volume 2013, Article number: 226 (2013)
Abstract
In this paper, based on the quadratic kernel function with three sections, which was defined by Liu in 2009, we establish some companions of perturbed Ostrowski-type inequalities for the case when , and , respectively. The special cases of these results offer better estimation than the conventional trapezoidal formula and the midpoint formula. The results we get can apply to composite quadrature rules in numerical integration and probability density functions. The effectiveness of these applications is also illustrated through several specific examples due to better error estimates.
MSC:26D15, 41A55, 41A80, 65C50.
1 Introduction
In 1938, Ostrowski [1] established the following interesting integral inequality for differentiable mappings with bounded derivatives.
Theorem 1.1 Let be a differentiable mapping on whose derivative is bounded on and denote . Then for all we have
The constant is sharp in the sense that it cannot be replaced by a smaller one.
In [2], Guessab and Schmeisser proved the following companion of Ostrowski’s inequality.
Theorem 1.2 Let be satisfying the Lipschitz condition, i.e., . Then for all we have
The constant is sharp in the sense that it cannot be replaced by a smaller one. In (1.2), the point gives the best estimator.
Motivated by [2], Dragomir [3] proved some companions of Ostrowski’s inequality for absolutely continuous functions. Recently, Alomari [4] studied the companion of Ostrowski inequality (1.2) for differentiable bounded mappings. In [5], Liu established some companions of an Ostrowski-type integral inequality for functions whose first derivatives are absolutely continuous and second derivatives belong to () spaces.
Theorem 1.3 Let be such that is absolutely continuous on and . Then for all we have
The constant is sharp in the sense that it cannot be replaced by a smaller one.
For other related results, the reader may refer to [6–26] and the references therein.
The main aim of this paper is to establish some companions of perturbed Ostrowski-type inequalities for the case when , and , respectively. For our purpose, we will use the quadratic kernel function with three sections (see (2.1) below) which was defined by Liu in [5]. The special cases of the results we get offer better estimation than the conventional trapezoidal formula and the midpoint formula. These results can apply to composite quadrature rules in numerical integration and probability density functions. The effectiveness of these applications is also illustrated through several specific examples due to better error estimates.
2 Main results
To prove our main results, we need the following lemmas.
Lemma 2.1 [5]
Let be such that is absolutely continuous on . Denote by the kernel given by
then the identity
holds.
2.1 The case when and is bounded
Theorem 2.1 Let be such that is absolutely continuous on . If and , , then for all we have
and
where .
Proof From (2.2) and the facts
and
it follows that
We denote
If is an arbitrary constant, then we have
Furthermore, we have
To compute
we denote
If we choose , then we get . If we choose , then we get . A direct computation gives that
Therefore, we get
We also have
and
Therefore, we obtain (2.3) and (2.4) by using (2.7)-(2.10), (2.1)-(2.15) and choosing and in (2.10), respectively. □
Corollary 2.1 Under the assumptions of Theorem 2.1, choose
-
(1)
, we have
(2.16)(2.17) -
(2)
, we have
-
(3)
, we have
Corollary 2.2 Let f be as in Theorem 2.1. Additionally, if f is symmetric about , then for all we have
and
2.2 The case when
Theorem 2.2 Let be a thrice continuously differentiable mapping in with . Then for all we have
Proof Let be defined by (2.8). From (2.7), we get
If we choose in (2.9) and use the Cauchy inequality, then we get
We can use the Diaz-Metcalf inequality (see [[19], p.83] or [[25], p.424]) to get
We also have
Therefore, using the above relations, we obtain (2.18). □
Corollary 2.3 Under the assumptions of Theorem 2.2, choose
-
(1)
, we have
(2.22) -
(2)
, we have
-
(3)
, we have
Corollary 2.4 Let f be as in Theorem 2.1. Additionally, if f is symmetric about , i.e., , then for all we have
2.3 The case when
Theorem 2.3 Let be such that is absolutely continuous on with . Then for all we have
where is defined by
and S is defined in Theorem 2.1.
Proof Let be defined by (2.8). If we choose in (2.9) and use the Cauchy inequality and (2.21), then we get
□
Corollary 2.5 Under the assumptions of Theorem 2.3, choose
-
(1)
, we have
(2.25) -
(2)
, we have
-
(3)
, we have
Corollary 2.6 Let f be as in Theorem 2.1. Additionally, if f is symmetric about , then for all we have
3 Application to composite quadrature rules
Let be a partition of the interval and ().
Consider the perturbed composite quadrature rules
and
The following results hold.
Theorem 3.1 Let be such that is absolutely continuous on . If and , , then for all we have
where is defined by formula (3.1), and the remainder satisfies the estimate
and
Proof Applying inequality (2.1) and (2.1) to the intervals , we get
and
for . Now summing over i from 0 to and using the triangle inequality, we get (3.3) and (3.4). □
Theorem 3.2 Let be a thrice continuously differentiable mapping in with . Then for all we have
where is defined by formula (3.1), and the remainder satisfies the estimate
Proof Applying inequality (2.3) to the intervals , we get
for . Now summing over i from 0 to and using the triangle inequality, we get (3.5). □
Theorem 3.3 Let be such that is absolutely continuous on with . Then for all we have
where is defined by formula (3.1), and the remainder satisfies the estimate
Proof Applying inequality (2.5) to the intervals , we get
for . Now summing over i from 0 to and using the triangle inequality, we get (3.6). □
To illustrate the effectiveness of the perturbed composite quadrature rules (3.1) and (3.2), we compute the approximate values of several specific examples using these two rules and the composite trapezoidal formula
respectively, and then we compare their errors. We get Table 1, from which the power of these two rules in numerical integration is demonstrated due to better error estimates.
4 Application to probability density functions
Now, let X be a random variable taking values in the finite interval , with the probability density function and with the cumulative distribution function
The following results hold.
Theorem 4.1 With the assumptions of Theorem 2.1, we have
and
for all , where is the expectation of X.
Proof By (2.3) and (2.4) on choosing and taking into account
we obtain (4.1) and (4.2). □
Corollary 4.1 Under the assumptions of Theorem 4.1 with , we have
and
Theorem 4.2 With the assumptions of Theorem 2.2, we have
for all , where is the expectation of X.
Proof By (2.18) on choosing and taking into account
we obtain (4.5). □
Corollary 4.2 Under the assumptions of Theorem 4.2 with , we have
Theorem 4.3 With the assumptions of Theorem 2.3, we have
for all , where is the expectation of X.
Proof By (2.23) on choosing and taking into account
we obtain (4.7). □
Corollary 4.3 Under the assumptions of Theorem 4.3 with , we have
References
Ostrowski A: Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert. Comment. Math. Helv. 1937, 10(1):226–227. 10.1007/BF01214290
Guessab A, Schmeisser G: Sharp integral inequalities of the Hermite-Hadamard type. J. Approx. Theory 2002, 115(2):260–288. 10.1006/jath.2001.3658
Dragomir SS: Some companions of Ostrowski’s inequality for absolutely continuous functions and applications. Bull. Korean Math. Soc. 2005, 42(2):213–230.
Alomari MW: A companion of Ostrowski’s inequality with applications. Transylv. J. Math. Mech. 2011, 3(1):9–14.
Liu Z: Some companions of an Ostrowski type inequality and applications. JIPAM. J. Inequal. Pure Appl. Math. 2009., 10(2): Article ID 52
Alomari MW: A companion of Ostrowski’s inequality for mappings whose first derivatives are bounded and applications in numerical integration. Transylv. J. Math. Mech. 2012, 4(2):103–109.
Alomari MW: A generalization of companion inequality of Ostrowski’s type for mappings whose first derivatives are bounded and applications in numerical integration. Kragujev. J. Math. 2012, 36(1):77–82.
Barnett NS, Dragomir SS, Gomm I: A companion for the Ostrowski and the generalised trapezoid inequalities. Math. Comput. Model. 2009, 50(1–2):179–187. 10.1016/j.mcm.2009.04.005
Dragomir SS: A companions of Ostrowski’s inequality for functions of bounded variation and applications. RGMIA Res. Rep. Coll. 2002., 5: Article ID 28
Dragomir SS: Ostrowski type inequalities for functions defined on linear spaces and applications for semi-inner products. J. Concr. Appl. Math. 2005, 3(1):91–103.
Dragomir SS: Ostrowski’s type inequalities for continuous functions of selfadjoint operators on Hilbert spaces: a survey of recent results. Ann. Funct. Anal. 2011, 2(1):139–205.
Dragomir SS, Sofo A: An integral inequality for twice differentiable mappings and applications. Tamkang J. Math. 2000, 31(4):257–266.
Duoandikoetxea J: A unified approach to several inequalities involving functions and derivatives. Czechoslov. Math. J. 2001, 51(126)(2):363–376.
Huy VN, Ngô Q-A: New bounds for the Ostrowski-like type inequalities. Bull. Korean Math. Soc. 2011, 48(1):95–104.
Liu WJ: Several error inequalities for a quadrature formula with a parameter and applications. Comput. Math. Appl. 2008, 56(7):1766–1772. 10.1016/j.camwa.2008.04.016
Liu WJ: Some weighted integral inequalities with a parameter and applications. Acta Appl. Math. 2010, 109(2):389–400. 10.1007/s10440-008-9323-2
Liu WJ, Xue QL, Wang SF: New generalization of perturbed Ostrowski type inequalities and applications. J. Appl. Math. Comput. 2010, 32(1):157–169. 10.1007/s12190-009-0240-y
Liu Z: Note on a paper by N. Ujević. Appl. Math. Lett. 2007, 20(6):659–663. 10.1016/j.aml.2006.09.001
Mitrinović DS, Pečarić JE, Fink AM: Inequalities involving functions and their integrals and derivatives. 53. In Mathematics and Its Applications (East European Series). Kluwer Academic, Dordrecht; 1991.
Sarikaya MZ: On the Ostrowski type integral inequality. Acta Math. Univ. Comen. 2010, 79(1):129–134.
Sarikaya MZ: New weighted Ostrowski and čebyšev type inequalities on time scales. Comput. Math. Appl. 2010, 60(5):1510–1514. 10.1016/j.camwa.2010.06.033
Set E, Sarıkaya MZ: On the generalization of Ostrowski and grüss type discrete inequalities. Comput. Math. Appl. 2011, 62(1):455–461. 10.1016/j.camwa.2011.05.026
Tseng K-L, Hwang S-R, Dragomir SS: Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications. Comput. Math. Appl. 2008, 55(8):1785–1793. 10.1016/j.camwa.2007.07.004
Tseng K-L, Hwang S-R, Yang G-S, Chou Y-M: Weighted Ostrowski integral inequality for mappings of bounded variation. Taiwan. J. Math. 2011, 15(2):573–585.
Ujević N: New bounds for the first inequality of Ostrowski-grüss type and applications. Comput. Math. Appl. 2003, 46(2–3):421–427. 10.1016/S0898-1221(03)90035-6
Vong SW: A note on some Ostrowski-like type inequalities. Comput. Math. Appl. 2011, 62(1):532–535. 10.1016/j.camwa.2011.05.037
Acknowledgements
The authors wish to thank the anonymous referees and the editor for their valuable comments. This work was partly supported by the Qing Lan Project of Jiangsu Province, the National Natural Science Foundation of China (Grant No. 41174165) and the Teaching Research Project of NUIST (Grant No. 12JY052).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to the writing of the present article and they read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Liu, W., Zhu, Y. & Park, J. Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications. J Inequal Appl 2013, 226 (2013). https://doi.org/10.1186/1029-242X-2013-226
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-226