Open Access

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

Journal of Inequalities and Applications20132013:1

DOI: 10.1186/1029-242X-2013-1

Received: 11 June 2012

Accepted: 28 November 2012

Published: 3 January 2013

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 [35]. 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 [35]. Moreover, results investigated [6] were in weighted form instead of previous results [35] 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 [35]. 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,
| f ( x ) 1 b a a b f ( t ) d t | [ 1 4 + ( x a + b 2 b a ) 2 ] ( b a ) f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ1_HTML.gif
(1)

for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif, where f L ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq2_HTML.gif and f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq3_HTML.gif is a differentiable mapping on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq4_HTML.gif.

The constant 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq5_HTML.gif is sharp in the sense that it cannot be replaced by a smaller one. We also observe that the tightest bound is obtained at x = a + b 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq6_HTML.gif, 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 1 Let f , g : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq7_HTML.gif be integrable functions such that φ f ( x ) Φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq8_HTML.gif and γ g ( x ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq9_HTML.gif for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif, where φ, Φ, γ, Γ are constants. Then
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ2_HTML.gif
(2)

where the constant 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq5_HTML.gif is sharp.

During the past few years, many researchers [710] 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 2 Let f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq10_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq11_HTML.gif and differentiable on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq4_HTML.gif, and its derivative satisfy the condition γ f ( x ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq12_HTML.gif for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq13_HTML.gif. Then we have the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ3_HTML.gif
(3)

for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif.

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 3 Let f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq14_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq11_HTML.gif and twice differentiable on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq4_HTML.gif, and assume that the second derivative f : ( a , b ) R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq15_HTML.gif satisfies the condition: γ f ( x ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq16_HTML.gif for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif.

Then, for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif, we have the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ4_HTML.gif
(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) w : ( a , b ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq17_HTML.gif to be non-negative and integrable over its entire domain and consider a b w ( t ) d t < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq18_HTML.gif. We denote the moments to be m, M and σ and define them as follows: m ( a , b ) = a b w ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq19_HTML.gif, M ( a , b ) = a b t w ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq20_HTML.gif and σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq21_HTML.gif. We start with the following weighted Grüss inequality [12].

Theorem 4 Let f , g : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq22_HTML.gif be two integrable functions such that θ f ( x ) ϕ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq23_HTML.gif and γ g ( x ) Γ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq24_HTML.gif for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif, and let ϕ, θ, Γ, γ be constants. Then we have | 1 m ( a , b ) a b w ( x ) f ( x ) g ( x ) d x 1 m ( a , b ) a b w ( x ) f ( x ) d x × 1 m ( a , b ) a b w ( x ) g ( x ) d x | 1 4 ( ϕ θ ) ( Γ γ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq25_HTML.gif, the constant 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq5_HTML.gif is sharp.

Now, we give our main result.

Theorem 5 Let f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq10_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq11_HTML.gif and differentiable on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq26_HTML.gif and f L 1 ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq27_HTML.gif. Then, for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq28_HTML.gif, we have the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ5_HTML.gif
(5)
Proof The following weighted integral inequality for all x [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq1_HTML.gif is proved in [13].
f ( x ) = 1 m ( a , b ) a b P ( x , t ) f ( t ) d t + 1 m ( a , b ) a b f ( t ) w ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ6_HTML.gif
(6)
where the weighted Peano kernel, P ( , ) : [ a , b ] 2 R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq29_HTML.gif, is given by
P ( x , t ) = { a t w ( u ) d u , if  t [ a , x ] , b t w ( u ) d u , if  t ( x , b ] , where  t [ a , b ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ7_HTML.gif
(7)
We observe that the mapping P ( , ) : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq30_HTML.gif satisfies the estimation
0 P ( x , t ) { x b w ( u ) d u , if  x [ a , a + b 2 ) a x w ( u ) d u , if  x [ a + b 2 , b ] } . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ8_HTML.gif
(8)
Consider, f ( x ) = p ( x , t ) w ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq31_HTML.gif and g ( x ) = f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq32_HTML.gif. Applying the weighted Grüss inequality to P ( x , t ) w ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq33_HTML.gif and f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq34_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ9_HTML.gif
(9)
Now, from (7), it can be easily seen that a b P ( x , t ) = m ( a , b ) ( x σ ( a , b ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq35_HTML.gif. Thus, (13) gives
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ10_HTML.gif
(10)
Using (6), the inequality (10) gives
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ11_HTML.gif
(11)
Further, we observe that
max ( x b w ( u ) d u , a x w ( u ) d u ) = { x b w ( u ) d u , if  x [ a , a + b 2 ) a x w ( u ) d u , if  x [ a + b 2 , b ] } = 1 2 m ( a , b ) + 1 2 | a b sgn ( t x ) w ( t ) d t | . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ12_HTML.gif
(12)

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

Corollary 6 Under the assumptions of Theorem 5 and choosing x = a + b 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq6_HTML.gif, we have the perturbed midpoint inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ13_HTML.gif
(13)

Proof This follows by inequality (5). □

Corollary 7 Under the assumptions of Theorem 5, we have the perturbed trapezoidal inequality
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ14_HTML.gif
(14)

Proof Put x = a https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq36_HTML.gif and x = b https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq37_HTML.gif 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 w ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq38_HTML.gif into the moment σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq21_HTML.gif gives σ ( a , b ) = a + b 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq39_HTML.gif. Substituting it into (5) gives
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ15_HTML.gif
(15)

Note that the interval mean σ ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq40_HTML.gif is simply the midpoint.

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

Putting w ( t ) = ln 1 t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq41_HTML.gif, a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq42_HTML.gif, b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq43_HTML.gif, the moment σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq21_HTML.gif and (5) imply
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ16_HTML.gif
(16)
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ17_HTML.gif
(17)

The optimal point σ ( 0 , 1 ) = 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq44_HTML.gif is closer to the origin than the midpoint σ ( a , b ) = a + b 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq45_HTML.gif, reflecting the strength of the log singularity.

Jacobi Substituting w ( t ) = 1 t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq46_HTML.gif, a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq42_HTML.gif, b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq43_HTML.gif, into the moment σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq21_HTML.gif gives
σ ( 0 , 1 ) = 0 1 t 1 t d t 0 1 1 t d t = 1 3 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ18_HTML.gif
(18)

Inequality (5) gives | f ( x ) 1 2 0 1 f ( t ) 1 t d t ( x 1 3 ) f ( x ) | 1 4 ( ϕ θ ) ( 1 + 1 2 | 0 1 sgn ( t x ) 1 t d t | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq47_HTML.gif.

The optimal point σ ( 0 , 1 ) = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq48_HTML.gif is again shifted to the left of the midpoint due to the 1 t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq49_HTML.gif singularity at the origin.

Chebyshev Substituting w ( t ) = 1 1 t 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq50_HTML.gif, a = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq51_HTML.gif, b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq43_HTML.gif, into the moment σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq21_HTML.gif gives σ ( 1 , 1 ) = 1 1 t 1 1 t 2 d t 1 1 1 1 t 2 d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq52_HTML.gif.

Hence, the inequality corresponding to the Chebyshev weight is | f ( x ) 1 m ( a , b ) 1 1 f ( t ) × 1 1 t 2 d t x f ( x ) | 1 4 ( ϕ θ ) ( π 2 + 1 2 | 1 1 sgn ( t x ) 1 1 t 2 d t | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq53_HTML.gif.

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

Laguerre The Laguerre weight w ( t ) = e t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq54_HTML.gif, is defined for positive values, t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq55_HTML.gif. From the moment σ ( a , b ) = M ( a , b ) m ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq56_HTML.gif, we have σ ( 0 , ) = 0 t e t d t 0 e t d t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq57_HTML.gif.

The appropriate inequality is | f ( x ) 0 f ( t ) e t d t ( x 1 ) f ( x ) | 1 4 ( ϕ θ ) ( 1 2 + 1 2 | 0 sgn ( t x ) e t d t | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq58_HTML.gif, from which the optimal sample point of x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq59_HTML.gif may be deduced.

Hermite Finally, the Hermite weight is w ( t ) = e t 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq60_HTML.gif defined over the entire real line σ ( , ) = t e t 2 d t e t 2 d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq61_HTML.gif. The inequality (5) with the Hermite weight function is thus | f ( x ) 1 π f ( t ) e t 2 d t x f ( x ) | 1 4 ( ϕ θ ) ( π 2 + 1 2 | sgn ( t x ) e t 2 d t | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq62_HTML.gif, which results in an optimal sampling point of x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq63_HTML.gif.

5 Application in numerical integration

Let I n : a = x 0 < x 1 < x 2 < < x n 1 < x n = b https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq64_HTML.gif be a division of the interval [ a , b ] , ξ i [ x i , x i + 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq65_HTML.gif ( i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq66_HTML.gif). We have the following quadrature formula.

Theorem 8 Let f : [ a , b ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq10_HTML.gif be continuous on [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq11_HTML.gif and differentiable on ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq26_HTML.gif, and f : ( a , b ) R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq67_HTML.gif satisfy the condition φ f ( x ) Φ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq68_HTML.gif for all x ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq69_HTML.gif. Then we have the following perturbed Riemann type quadrature formula: a b f ( t ) w ( t ) d t = A ( f , f , ξ , I n ) + R ( f , f , ξ , I n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq70_HTML.gif, where A ( f , f , ξ , I n ) = i = 0 n 1 m ( x i , x i + 1 ) f ( ξ i ) i = 0 n 1 m ( x i , x i + 1 ) ( x σ ( x i , x i + 1 ) ) f ( ξ i ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq71_HTML.gif and the remainder satisfies the estimation
R ( f , ξ , I n ) 1 8 ( ϕ θ ) i = 0 n 1 m ( x i , x i + 1 ) × ( m ( x i , x i + 1 ) + x i x i + 1 sgn ( t ξ i ) w ( t ) d t ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_Equ19_HTML.gif
(19)

for all ξ i [ x i , x i + 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq72_HTML.gif, where h i : = x i + 1 x i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq73_HTML.gif ( i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq66_HTML.gif).

Proof Apply Theorem 5 to the interval [ x i , x i + 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq74_HTML.gif, ξ i [ x i , x i + 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq72_HTML.gif, where h i : = x i + 1 x i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq73_HTML.gif ( i = 1 , 2 , , n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq66_HTML.gif), to get | x i x i + 1 f ( t ) w ( t ) d t + m ( x i , x i + 1 ) ( ξ i σ ( x i , x i + 1 ) ) f ( ξ i ) m ( x i , x i + 1 ) f ( ξ i ) | https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq75_HTML.gif 1 8 ( ϕ θ ) m ( x i , x i + 1 ) ( m ( x i , x i + 1 ) + x i x i + 1 sgn ( t ξ i ) w ( t ) d t ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq76_HTML.gif.

Summing over i from 0 to n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq77_HTML.gif and using the generalized triangular inequality, we deduce the desired estimation (19). □

Corollary 9 Under the assumption of Theorem 5, by choosing ξ i = x i + x i + 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq78_HTML.gif in the above theorem, we recapture the midpoint like quadrature formula: x i x i + 1 f ( t ) w ( t ) d t = A M ( f , f , ξ , I n ) + R M ( f , f , ξ , I n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq79_HTML.gif, where A M ( f , f , ξ , I n ) = i = 0 n 1 m ( x i , x i + 1 ) f ( x i + x i + 1 2 ) i = 0 n 1 m ( x i , x i + 1 ) ( x i + x i + 1 2 ) σ ( x i , x i + 1 ) f ( x i + x i + 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq80_HTML.gif, and the remainder term satisfies the estimation R M ( f , f , ξ , I n ) 1 8 ( Φ φ ) i = 0 n 1 ( m ( x i , x i + 1 ) + | x i x i + 1 sgn ( t x i + x i + 1 2 ) w ( t ) d t | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-1/MediaObjects/13660_2012_Article_449_IEq81_HTML.gif.

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 [35]. 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.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Yanbu University
(2)
Department of Mathematical Sciences, University of Hail

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