On New Generalized Ostrowski Type Integral Inequalities

Abstract

The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish some new inequalities similar to the Ostrowski's inequality. The current paper obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval in terms of the norms of the second derivative of the function. Some new perturbed results are obtained. Application for cumulative distribution function is also discussed.

1. Introduction

Ostrowski [1] proved his famous inequality in 1938. Milovanović and Pecarić gave the first generalization of Ostrowski's inequality [2]. A number of Ostrowski type inequalities have been derived by Cerone [3], Cheng [4], Dragomir and Barnett [5], Sofo and Dragomir [6], Milovanović [7], and G. V. Milovanović and I. Z. Milovanović [8] with applications in numerical analysis and probability. Several further generalizations are provided by Hussain and Qayyum [9], Qayyum and Hussain [10], and Pecarić [11], respectively. More recent results concerning the generalizations of Ostrowski inequality are given by [4, 5, 12, 13]. Dragomir and Wang [14] combined Ostrowski and Grüss inequality to give a new inequality which they named Ostrowski-Grüss type inequality. Cheng gave a sharp version of Ostrowski-Grüss inequality in [4]. Cerone [3] and Dragomir and Wang in [14–17] obtained bounds on a particular quadrature rule for differentiable functions which we generalize to two times differentiable functions. This generalization improves bounds on the deviation of a function from a combination of integral means. Some new perturbed results are also discussed. The generalized inequalities obtained in this paper will have applications in approximation theory, probability theory, and numerical analysis. We will show an application of the inequalities obtained for cumulative distribution function.

In this paper, we will use the usual norms defined for a function as follows: Let be defined by where is the integral mean of over . The functional represents the deviation of from its integral mean over .

Ostrowski [1] proved the following interesting and useful integral inequality.

Theorem 1. Let be continuous on and differentiable on , whose derivative is bounded on by a constant . Then, for all .

In a series of papers, Dragomir and Wang [14–17] proved (4) and some of its variants for , when , making use of a Peano kernel and Montgomery's identity [18]. Montgomery's identity states that for absolutely continuous mappings where the kernel is given by If we assume that , then in (4) may be replaced by .

Dragomir and Wang [14–17], utilizing an integration by parts argument, obtained where is absolutely continuous on and the constants , and are sharp.

Cerone [3] proved the following inequality.

Theorem 2. Let be absolutely continuous. Define Then,

In [19], Pachpatte established Čebyŝev type inequalities by using Pecarić's extension of the Montgomery identity [18]. The current paper obtains bounds on the deviation of a function from integral means from the end of the interval which cover the whole interval, in terms of the norms of the second derivative of the function. The paper closely follows ideas from [3], where corresponding results were proved using bounds in terms of the first derivative of the function.

2. Main Results

Denote by the kernel given by where are nonnegative and not both zero.

Before we state and prove our main result, we will prove the following identity, which will be used to obtain bounds.

Lemma 3. Let be two times differentiable on . Then,

Proof. From (10), we have After further simplification, we get the required identity (11).

We now give our main result.

Theorem 4. Let be a two times differentiable function. Define where is the integral mean defined in (3). Then, for all .

Proof. Taking the modulus of (11) and using (13) and (3), we have
Therefore, for , we obtain Now let us observe that Hence, the first inequality is obtained: Further, using Hölder's integral inequality, from (15), we have for , , where . Now Hence, the second inequality is obtained as below: Finally, for , using (10), we have the following from (15). Consider where This gives us the last inequality as below: This completes the proof of the theorem.

Remark 5. We may write where Thus, from (13), so that, for fixed , is also fixed.

Corollary 6. Let the condition of Theorem 4 hold and ; then,

Proof. The proof of the above corollary is a straightforward exercise after putting in (14).

Corollary 7. Let the conditions of Theorem 4 hold and . Then,

Proof. Placing in (13) and (14) produces the results stated in (29).

Corollary 8. If (28) is evaluated at the midpoint, then The above result can also be obtained by taking in (29) or equivalently and in (14).

3. Perturbed Results

In 1882, Čebyŝev [20] gave the following inequality: where are absolutely continuous functions, which have bounded first derivatives, and

In 1935, Grüss [21] proved the following inequality: provided that and are two integrable functions on and satisfy the condition The constant is the best possible. The perturbed version of the results of Theorem 4 can be obtained by using Grüss type results involving the Čebyŝev functional where is the integral mean defined in (3). Let

Theorem 9. Let be a two times differentiable function, and , are nonnegative real numbers. Then, where is as given by (13).

Proof. Associating with and with , we obtain Now using identity (11), where is the secant slope of over as given in (36). Now, from (11) and (32), Now, combining (41) with (37), the left-hand side of (38) is obtained.
Let and be integrable on ; then [3],
Also, note that where , . Now, for the bounds on (40), we have to determine and and such that .
Now from (10), the definition of , we have From (41) we obtain Thus, substituting the above results into (44) gives which is given explicitly by (37). Combining (40), (43), and (44) gives, from the first inequality in (42), the first inequality in (38). Now utilizing the inequality in (43) produces the second result in (38). Further, it may be noticed from the definition of in (10) that, for it gives, for the values

4. An Application to the Cumulative Distribution Function

Let be a random variable with the cumulative distributive function where is the probability density function. In particular, The following theorem holds.

Theorem 10. Let and be as above; then,

Proof. From (13), and by using the definition of probability density function, we have
or
Now using (14) and (53), we get our required result (51).

Putting in Theorem 10 gives the following result.

Corollary 11. Let be a random variable with the cumulative distributive function and the probability density function . Then,

Remark 12. The above result allows 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 13. We put in (51), assuming that , to obtain Further we note that

5. Conclusions

In this paper, we obtained bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval in terms of the norms at the second derivative. The obtained results are applied to approximate the cumulative distribution function. Our results further extend previous inequalities developed in [3, 14–17]. Some perturbed results with the help of Čebyŝev and Grüss inequalities are also obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank Professor Elena berdysheva for her useful suggestions which improved the presentation of the paper.