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 Ostrowski-Grüss 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 non-weighted version.

Lemma 3 Denote by : the kernel is given by

(10)

Then,

(11)

Qayyum et.al ^[15] proved a generalized non-weighted 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 non-negative 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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