SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications

Sabir Hussain1* and Ather Qayyum2

Author Affiliations

1 Department of Mathematics, Yanbu University, P.O. Box 31387, Yanbu Al Sinaiyah, Saudi Arabia

2 Department of Mathematical Sciences, University of Hail, P.O. Box 2440, Hail, Saudi Arabia

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:1  doi:10.1186/1029-242X-2013-1


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/1


Received:11 June 2012
Accepted:28 November 2012
Published:3 January 2013

© 2013 Hussain and Qayyum; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we establish a generalized Ostrowski-Grüss type inequality for differentiable mappings using the weighted Grüss inequality which is another generalization of inequalities established and discussed by Barnett et al. (Inequality theory and applications, pp. 24-30, 2001), S. S. Dragomir and S. Wang (Comput. Math. Appl. 33:15-22, 1997) and A. Rafiq et al. (JIPAM. J. Inequal. Pure Appl. Math. 7(4):124, 2006). Perturbed midpoint and trapezoid inequalities are obtained. Some applications in different weights are given. This inequality is extended to account for applications in numerical integration.

Keywords:
Ostrowski inequality; Grüss inequality; weight function; numerical integration

1 Introduction

Integration with weight functions is used in countless mathematical problems such as approximation theory and spectral analysis, statistical analysis and the theory of distributions. Grüss developed an integral inequality [1] in 1935. In 1938, Ostrowski [2] established an interesting integral inequality associated with differentiable mappings which has powerful applications in numerical integration, probability and optimization theory, stochastic, statistics, information and integral operator theory. During the last few years, many researchers focused their attention on the study and generalizations of the above two inequalities [3-5]. Recently, Qayyum and Hussain [6] established a new inequality using the weighted Peano kernel, which is more generalized as compared to previous inequalities developed and discussed in [3-5]. Moreover, results investigated [6] were in weighted form instead of previous results [3-5] which were in non-weighted form. This approach not only generalized the results of [3], but also gave some other interesting inequalities as special cases. In this paper, we establish another generalization of the Ostrowski-Grüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [3-5]. Perturbed midpoint and trapezoid inequalities are also obtained. In Section 4, we give some applications in different weights. This inequality is extended to account for applications in numerical integration in Section 5.

2 Preliminaries

The classical Ostrowski integral inequality ([2] see also [[1], p.468]) in one dimension stipulates a bound between a function evaluated at an interior point x and the average of the function f over an interval. That is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M1">View MathML</a>

(1)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M3">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M4">View MathML</a> is a differentiable mapping on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5">View MathML</a>.

The constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6">View MathML</a> is sharp in the sense that it cannot be replaced by a smaller one. We also observe that the tightest bound is obtained at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M7">View MathML</a>, resulting in the well-known mid-point inequality.

The integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals is known in the literature as the Grüss inequality. The inequality is as follows.

Theorem 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M8">View MathML</a>be integrable functions such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M9">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M10">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>, whereφ, Φ, γ, Γ are constants. Then

(2)

where the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6">View MathML</a>is sharp.

During the past few years, many researchers [7-10] have given considerable attention to the inequality (1).

In [4], Dragomir and Wang improved the above inequality and proved the following Ostrowski type inequality in terms of the lower and upper bounds of the first derivative.

Theorem 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15">View MathML</a>and differentiable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5">View MathML</a>, and its derivative satisfy the condition<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M17">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M18">View MathML</a>. Then we have the inequality

(3)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>.

In [3], Barnett et al. pointed out a similar result to the above for twice differentiable mappings in terms of the upper and lower bounds of the second derivative.

Theorem 3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M21">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15">View MathML</a>and twice differentiable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M5">View MathML</a>, and assume that the second derivative<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M24">View MathML</a>satisfies the condition: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M25">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>.

Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>, we have the inequality

(4)

In the recent years, some authors (see, for example, [5,6,11]) also generalized the above inequality.

3 Some new results

We assume the weight function (or density) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M29">View MathML</a> to be non-negative and integrable over its entire domain and consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M30">View MathML</a>. We denote the moments to be m, M and σ and define them as follows: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M31">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M32">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33">View MathML</a>. We start with the following weighted Grüss inequality [12].

Theorem 4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M34">View MathML</a>be two integrable functions such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M35">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M36">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a>, and letϕ, θ, Γ, γbe constants. Then we have<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M38">View MathML</a>, the constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M6">View MathML</a>is sharp.

Now, we give our main result.

Theorem 5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15">View MathML</a>and differentiable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M42">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M43">View MathML</a>. Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M44">View MathML</a>, we have the inequality

(5)

Proof The following weighted integral inequality for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M2">View MathML</a> is proved in [13].

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M47">View MathML</a>

(6)

where the weighted Peano kernel, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M48">View MathML</a>, is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M49">View MathML</a>

(7)

We observe that the mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M50">View MathML</a> satisfies the estimation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M51">View MathML</a>

(8)

Consider, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M52">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M53">View MathML</a>. Applying the weighted Grüss inequality to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M54">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M55">View MathML</a>, we get

(9)

Now, from (7), it can be easily seen that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M57">View MathML</a>. Thus, (13) gives

(10)

Using (6), the inequality (10) gives

(11)

Further, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M60">View MathML</a>

(12)

Using (12) in (11), we get our main result (5). □

Corollary 6Under the assumptions of Theorem 5 and choosing<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M7">View MathML</a>, we have the perturbed midpoint inequality

(13)

Proof This follows by inequality (5). □

Corollary 7Under the assumptions of Theorem 5, we have the perturbed trapezoidal inequality

(14)

Proof Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M64">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M65">View MathML</a> in (5) and sum up the obtained inequalities. Using the triangle inequality and dividing by two, we get the required inequality. □

4 Some weighted integral inequalities

Integration with weight functions is used in countless mathematical problems. Two main areas are: (i) approximation theory and spectral analysis and (ii) statistical analysis and the theory of distributions. In this section, inequality (5) is evaluated for the more popular weight functions.

Uniform (Legender) Substituting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M66">View MathML</a> into the moment <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33">View MathML</a> gives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M68">View MathML</a>. Substituting it into (5) gives

(15)

Note that the interval mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M70">View MathML</a> is simply the midpoint.

Logarithm This weight is present in many physical problems, the main body of which exhibits some axial symmetry.

Putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M71">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M72">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73">View MathML</a>, the moment <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33">View MathML</a> and (5) imply

(16)

(17)

The optimal point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M77">View MathML</a> is closer to the origin than the midpoint <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M78">View MathML</a>, reflecting the strength of the log singularity.

Jacobi Substituting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M79">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M72">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73">View MathML</a>, into the moment <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33">View MathML</a> gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M83">View MathML</a>

(18)

Inequality (5) gives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M84">View MathML</a>.

The optimal point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M85">View MathML</a> is again shifted to the left of the midpoint due to the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M86">View MathML</a> singularity at the origin.

Chebyshev Substituting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M87">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M88">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M73">View MathML</a>, into the moment <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M33">View MathML</a> gives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M91">View MathML</a>.

Hence, the inequality corresponding to the Chebyshev weight is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M92">View MathML</a>.

The optimal point is at the midpoint of the interval reflecting the symmetry of the Chebyshev weight over its interval.

Laguerre The Laguerre weight <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M93">View MathML</a>, is defined for positive values, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M94">View MathML</a>. From the moment <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M95">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M96">View MathML</a>.

The appropriate inequality is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M97">View MathML</a>, from which the optimal sample point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M98">View MathML</a> may be deduced.

Hermite Finally, the Hermite weight is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M99">View MathML</a> defined over the entire real line <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M100">View MathML</a>. The inequality (5) with the Hermite weight function is thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M101">View MathML</a>, which results in an optimal sampling point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M102">View MathML</a>.

5 Application in numerical integration

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M103">View MathML</a> be a division of the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M104">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105">View MathML</a>). We have the following quadrature formula.

Theorem 8Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M14">View MathML</a>be continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M15">View MathML</a>and differentiable on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M42">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M109">View MathML</a>satisfy the condition<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M110">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M111">View MathML</a>. Then we have the following perturbed Riemann type quadrature formula: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M113">View MathML</a>and the remainder satisfies the estimation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M114">View MathML</a>

(19)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M115">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M116">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105">View MathML</a>).

Proof Apply Theorem 5 to the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M118">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M115">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M116">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M105">View MathML</a>), to get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M122">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M123">View MathML</a>.

Summing over i from 0 to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M124">View MathML</a> and using the generalized triangular inequality, we deduce the desired estimation (19). □

Corollary 9Under the assumption of Theorem 5, by choosing<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M125">View MathML</a>in the above theorem, we recapture the midpoint like quadrature formula: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M126">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M127">View MathML</a>, and the remainder term satisfies the estimation<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M128">View MathML</a>.

6 Conclusion

We established another generalization of the Ostrowski-Grüss type inequality using the weighted Grüss inequality for bounded differentiable mappings which generalizes the previous inequalities developed and discussed in [3-5]. Perturbed midpoint and trapezoid inequalities are also obtained. This inequality is extended to account for applications in different weights and numerical integration. This generalized inequality will be useful for the researchers working in the field of the numerical analysis to solve their problems in engineering and in practical life.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgements

The first author acknowledge the financial support from the Research and Development Center of Colleges and Institute of Royal Commission at Yanbu for this research.

References

  1. Mitrinović, DS, Pěcarić, JE, Fink, AM: Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht (1993)

  2. Ostrowski, AM: Über die absolutabweichung einer differentiebaren funktion von ihrem integralmittelwert. Comment. Math. Helv.. 10, 226–227 (1938)

  3. Barnett, NS, Cerone, P, Dragomir, SS, Roumeliotis, J, Sofo, A: A survey on Ostrowski type inequalities for twice differentiable mappings and applications. Inequality Theory and Applications, pp. 24–30. Nova Publ., Huntington (2001)

  4. Dragomir, SS, Wang, S: An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. Comput. Math. Appl.. 33, 15–22 (1997)

  5. Rafiq, A, Mir, NA, Zafar, F: A generalized Ostrowski-Grüss type inequality for twice differentiable mappings and applications. JIPAM. J. Inequal. Pure Appl. Math.. 7(4), Article ID 124 (2006)

  6. Qayyum, A, Hussain, S: A new generalized Ostrowski Grüss type inequality and applications. Appl. Math. Lett.. 25, 1875–1880 doi:10.1016/j.aml.2012.02.052 (2012)

    doi:10.1016/j.aml.2012.02.052

    Publisher Full Text OpenURL

  7. Milovanovic, GV, Pĕcairć, JE: On a generalization of the inequality of Ostrowski and some related applications. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz.. 544-576, 155–158 (1976)

  8. Barnett, NS, Cerone, P, Dragomir, SS, Roumeliotis, J, Sofo, A: A survey on Ostrowski type inequalities for twice differentiable mappings and applications. Inequality Theory and Applications, pp. 24–30. Nova Publ., Huntington (2001)

  9. Dragomir, SS, Sofo, A: Trapezoidal type inequalities for n-type differentiable functions. RGMIA, Research Report Collection 8(3), (2005)

  10. Fink, AM: Bounds on the deviation of a function from its averages. Czechoslov. Math. J.. 42(117), 280–310 (1992)

  11. Rafiq, A, Mir, NA, Ahmad, F: Weighted Ostrowski type inequality for differentiable mappings whose first derivatives belong to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/1/mathml/M129">View MathML</a>. Gen. Math.. 14(3), 91–102 (2006)

  12. Dragomir, SS: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math.. 31(4), 397–415 (2000)

  13. Qayyum, A: A weighted Ostrowski-Grüss type inequality for twice differentiable mappings and applications. Int. J. Math. Comput. Sci. (Print). 1(8), 63–71 (2008)