Keywords: ostrowski inequality, weight function, weighted integral mean
Turkish Journal of Analysis and Number Theory, 2015 3 (2),
pp 6167.
DOI: 10.12691/tjant325
Received October 10, 2014; Revised February 08, 2015; Accepted May 20, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
Integration with weight functions is used in many mathematical branches such as: approximation theory, spectral analysis, statistical analysis, theory of distributions and functional analysis. In 1938, Ostrowski ^{[13]} proved his famous inequality:
Theorem 1 Let : be continuous on and differentiable on and assume for all , then the inequality holds
 (1) 
for all . The constant is best possible.
Several generalizations of the above integral inequality are considered by many authors such as ^{[2, 3, 4, 10, 16, 17]}.
The functional given below, represents the deviation of from its integral mean over
 (2) 
and
 (3) 
Milovanovic’ and Pecaric’ in ^{[11]} gave first generalization of Ostrowski’s inequality. Dragomir and Wang ^{[7]} introduced OstrowskiGrüss type inequality. Cheng gave a sharp version of the mentioned inequality in ^{[3]}. Dragomir and Wang ^{[5, 6, 7, 8, 9]} and Cerone ^{[1]} pointed out a result to the above. We establish new inequalities, which are more generalized as compared to previous inequalities developed in ^{[1, 5, 6, 7, 8, 9]} and ^{[14, 15]}. Moreover, our results are further generalized by the introduction of weight function.
The approach of Dragomir and Wang ^{[5, 6, 7, 8, 9]}, Cerone ^{[1]}, and Qayyum et. al ^{[14, 15]} for obtaining bounds depended on Peano kernel while we use weighted Peano kernel in our findings. This approach not only generalizes the results of Dragomir and Wang ^{[5, 6, 7, 8, 9]}, Cerone ^{[1]} and Qayyum ^{[14, 15]}, but also gives some other interesting inequalities as special cases. In the last section, we will apply our main result for cumulative distribution function.
Montgomery’s identity states that for absolutely continuous mappings :
 (4) 
where
Dragomir and Wang ^{[5, 6, 7, 8]} utilizing an integration by parts argument, obtained
 (5) 
where : is absolutely continuous on and the constants and are sharp. is defined in ^{[14]}.
Cerone ^{[1]}, proved the following inequality:
Lemma 1 Let : be absolutely continuous mapping and define
 (6) 
where
then
 (7) 
Now we express some generalizations (non weighted version) of cerone’s result ^{[1]}.
Lemma 2 Let the peano type kernel, is given by
 (8) 
Then,
 (9) 
Qayyum et.al ^{[14]} proved another nonweighted version.
Lemma 3 Denote by : the kernel is given by
 (10) 
Then,
 (11) 
Qayyum et.al ^{[15]} proved a generalized nonweighted version of Cerone’s result ^{[1]}.
Lemma 4 Let : be an absolutely continuous mapping. Denote by : the kernel is given by
Then,
 (12) 
2. Main Results
To establish our main results we first give the following essential definition and lemma.
Definition 1 We assume that the weight function (or density) to be nonnegative and integrable over its entire domain.
and
The domain of may be finite or infinite and may vanish at the boundary point. We denote the moments
Let the functional be defined by
 (13) 
where
 (14) 
The function represents the deviation of from its weighted integral mean over
We start our main result with the following weighted identity which will be used to obtain bounds.
Lemma 5 Let : be an absolutely continuous mapping. Denoted by : the weighted peano kernel is then given by
 (15) 
where and not both zero. Then the following weighted identity
 (16) 
holds.
Proof. From (15), we have
By integration by parts formula, we get (16).
We now give our main result.
Theorem 2 Let : be absolutely continuous mapping. Define
 (17) 
where is the weighted integral mean defined in (14) then
 (18) 
Proof. Taking the modulus of (16), we have from (14) and (17)
 (19) 
Thus, for (19) gives
Using (15) we obtain
Hence the first inequality is proved.
Further, using Hölder’s integral inequality, from (19) we have for
where with Now
This proves the second inequality.
Finally, for , we have from (19) and using (15)
where
This completes the proof of theorem.
Remark 1 If we put in (18) we get cerone’s results (7) If we put and in (18)we get Dragomir’s result (5) Similarly, for different weights, we can obtain a variety of results.
Remark 2 We may write
where
 (20) 
Thus, from (17),
so that for fixed is also fixed.
Corollary 1 If we put in (18), we get
 (21) 
Corollary 2 Let the conditions of Theorem 2 holds. Then
 (22) 
Proof. Placing in (17) and (18) produces the results stated in (22)
Corollary 3 If (21) is evaluated at the midpoint, then we get
 (23) 
3. An Application to the Cumulative Distribution Function
Let be a random variable taking values in the finite interval with Cumulative Distributive Function
we also use the fact that
where is a Probability Density Function. The following theorem holds.
Theorem 3 Let and be as above, then
 (24) 
Proof. From (18) we have
After simple calculations, we get
By using (18), we get (24).
Putting in Theorem 3 gives the following result.
Corollary 4 Let be a random variable, weighted Cumulative Distributive Function and is a Probability Density Function. Then
 (25) 
Remark 3 The above result allow the approximation of in terms of . The approximation of
could also be obtained by a simple substitution. is of importance in reliability theory where is the Probability Density Function of failure.
Remark 4 Put in (24) and, assuming that to obtain
 (26) 
Further we note that
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