On Pompeiu-Chebyshev functional and its generalization
Mohammad W. Alomari
TL;DR
This paper generalizes the Chebyshev functional, establishes new inequalities using Pompeiu's mean value theorem, and applies these results to bounds in inequalities like CBS and Hardy type inequalities.
Contribution
It introduces a broad generalization of the Chebyshev functional, derives new inequalities, and extends classical results with applications to reverse CBS and Hardy inequalities.
Findings
New inequalities of Gruss type established
Bounds for reverse CBS inequality derived
Hardy type inequalities on [a,b] introduced
Abstract
In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu's mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval [a,b] under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.
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On Pompeiu–Chebyshev functional and its generalization
Mohammad W. Alomari
Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600, Irbid, P.C. 21110, Jordan.
Abstract.
In this work, a generalization of Chebyshev functional is presented. New inequalities of Grüss type via Pompeiu’s mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Grüss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.
Key words and phrases:
Chebyshev functional, Grüss inequality, Pompeiu MVT, CBS inequality, Hardy inequality.
2010 Mathematics Subject Classification:
26D15, 26D10, 26A48
1. Introduction
The difference
[TABLE]
is called ‘the Chebyshev functional’ which it has multiple applications in several mathematical branches specially in Numerical integrations and Probability Theory. For more detailed history see [22].
The most famous bounds of the Chebyshev functional are incorporated in the following theorem:
Theorem 1**.**
Let be two absolutely continuous functions, then
[TABLE]
The constants , , and are the best possible.
The first inequality in \reftagform@1.9, is well known as Chebyshev inequality (sometimes called Chebyshev second inequality) which deals with differentiable functions whose first derivatives are bounded. The second inequality in \reftagform@1.9 is called the Grüss inequality which deals with integrable bounded functions.
Far away from this, Pompeiu in [26] established a variant Mean Value Theorem (MVT) for real functions defined on a real interval that does not include ‘[math]’; nowadays known as Pompeiu’s mean value theorem (PMVT), which states that:
Theorem 2**.**
For every real valued function differentiable on an interval not containing [math] and for all pairs in there exists a point such that
[TABLE]
The geometrical interpretation of this theorem as given in [28]: the tangent at the point intersects the -axis at the same point as the secant line connecting the points and . The proof of PMVT can be done by applying the generalized MVT for derivatives on the functions and , where is differentiable on .
In 2005, and in viewing of PMVT, Pachpatte [24] proposed the following corresponding Chebyshev functional: For continuous functions which are differentiable on , define
[TABLE]
and let us call it “Pompeiu–Chebyshev functional”. In [24], we find the following result:
Theorem 3**.**
Let . Let be two continuous functions on and differentiable on . Then
[TABLE]
where , for all .
In the same year, Pečarić and Ungar [25] studied the Pompeiu–Chebyshev functional \reftagform@1.11, and they obtained the following result.
Theorem 4**.**
Let the functions be continuous on and differentiable on with . Then for , with , the inequality
[TABLE]
holds, where , and
[TABLE]
for all .
Remark 1**.**
In particular, we have the following special cases in \reftagform@1.13:
[TABLE]
for all .
In 2015, Dragomir [9] studied the Pompeiu-Chebyshev functional \reftagform@1.11 with a slightly different consideration. Namely, Dragomir considered \reftagform@1.11 as
[TABLE]
The following lemma plays a main role in the proof of Dragomir results in [9].
Lemma 1**.**
Let be an absolutely continuous function on the interval with . Then for any , we have
[TABLE]
where with .
The main results in [9], are combined in the following theorem.
Theorem 1**.**
Let be absolutely continuous functions on the interval with .
- (1)
If , then
[TABLE]
The constant is the best possible. 2. (2)
If and , , and , then
[TABLE]
where
[TABLE] 3. (3)
If , then
[TABLE] 4. (4)
If , then
[TABLE]
where , .
Remark 1**.**
It is convenient to remark here that, Dragomir proved the sharpness of \reftagform@1.15 making use of the constant functions ; which is trivial case. It would be more useful if the sharpness holds for non-trivial functions. For this purpose, we consider given by , , and any arbitrary non-zero constant. Simple calculations yield the desired sharpness.
Some other inequalities were introduced in literature by many authors, for recent and related results we refer the reader to [1]–[3], [8]–[10], [20] and [27].
This work is divided into five sections, after this introduction, the second and third sections are devoted to elaborate and investigate some new inequalities of Grüss type via Pompeiu’s mean value theorem. Improvements of some old inequalities are also provided. In section 4, generalizations of Grüss type inequalities via Boggio mean value theorem are established. As applications, bounds for the reverse CBS inequality are obtained. In section 5, Using some extracted functionals; new Hardy type inequalities and their generalizations are detected. Some other inequalities for differentiable functions are also given.
2. The Results
Let us start with the following results regarding positivity of .
Theorem 2**.**
Let with . Let be two Lebesgue integrable functions on and satisfy the condition
[TABLE]
for all . Then
[TABLE]
Proof.
Let and be satisfied the given condition
[TABLE]
for all , therefore we have
[TABLE]
but also, we have
[TABLE]
which proves the inequality \reftagform@2.1. ∎
Corollary 1**.**
Let with . Let be two Lebesgue integrable functions on and satisfy the condition
[TABLE]
for all . Then
[TABLE]
Remark 2**.**
Let with . Let be an increasing on then the function given by is increasing. So that for any two distinct points with , we have . The reverse observation holds for decreasing function . A generalization of monotonicity and thus the previous two results are given in Section 4.
A pre-Grüss like inequality is incorporated in the following theorem:
Theorem 3**.**
Let with . Let be two Lebesgue integrable functions then the inequality
[TABLE]
holds and sharp.
Proof.
It is easy to verify that
[TABLE]
For instance, we observe that
[TABLE]
Now, using the triangle integral inequality and the Cauchy-Schwarz inequality, we have
[TABLE]
as desired. The sharpness follows by considering , , and any arbitrary non-zero constant. ∎
Theorem 4**.**
Let with . Let be two measurable functions. If there exists real numbers such that and for all , then the inequality
[TABLE]
holds.
Proof.
Since for all , then and adding the last two inequalities we get that or we may write . Similarly, for we have . So that we have
[TABLE]
which proves the result \reftagform@2.5. ∎
Theorem 5**.**
Let with . Let be a Lebesgue integrable and be a measurable and there exists real numbers such that for all , then the inequality
[TABLE]
holds.
Proof.
Since for all , then and adding the last two inequalities we get that or we may write .
Now since
[TABLE]
it follows that . Employing \reftagform@2.3 we get the required result. ∎
An improvement of \reftagform@2.5 is given in the following theorem.
Theorem 6**.**
Let with . Let be two measurable functions on . If there exists real numbers such that and for all , that satisfies the condition , then the inequality
[TABLE]
holds.
Proof.
Since for all , then , and , adding these two inequalities we get that . Similarly, for we have . So that we have
[TABLE]
Now, substituting and , where and . Solving the last two equation with respect to and we find that
[TABLE]
Clearly, the Jacobian , and thus we have
[TABLE]
To evaluate the integral in \reftagform@2.9, we have the following cases:
Case I: If and , then
[TABLE]
substituting \reftagform@2.10 in \reftagform@2.8 we get the first inequality in \reftagform@2.7.
Case II: If and , then
[TABLE]
substituting \reftagform@2.11 in \reftagform@2.8 we get the fourth inequality in \reftagform@2.7.
Case III: If and , then
[TABLE]
substituting \reftagform@2.12 in \reftagform@2.8 we get the third inequality in \reftagform@2.7.
Case IV: If and , then
[TABLE]
substituting \reftagform@2.13 in \reftagform@2.8 we get the second inequality in \reftagform@2.7.
∎
Remark 3**.**
Theorem 6 does not work for two identical functions. In other words, we cannot choose or a.e. on . The reason behind this is that, if one chooses then the substitution and , will be the same and so that we have and this contradicts the assumption that .
More general and extensive case can be done through the following improvement of \reftagform@2.6.
Theorem 7**.**
Under the assumptions of Theorem 5, then the inequality
[TABLE]
holds.
Proof.
From Theorem 3, we can state the following
[TABLE]
which gives the desired result in \reftagform@2.14. ∎
Remark 4**.**
In Theorem 7, if then
[TABLE]
So that, if there exist , for all , then
[TABLE]
which improves and generalizes \reftagform@2.7.
Next, an improvement of \reftagform@1.17 can be presented as follows:
Theorem 8**.**
Let with . Let be two absolutely continuous functions. If , then we have
[TABLE]
where , .
Proof.
Since we have
[TABLE]
Let us write
[TABLE]
In [21], G. Milovanović and Ž. Milovanović proved the following inequality:
[TABLE]
for any . where ia positive and continuous on with and is absolutely continuous on with and . The inequality is sharp.
In viewing of \reftagform@2.17 and by setting where and , with for all , with in \reftagform@2.18 we can state that
[TABLE]
Integrating w. r. to over , we get
[TABLE]
Similarly, for we have
[TABLE]
Substituting \reftagform@2.20 and \reftagform@2.21 in \reftagform@2.16 we get the desired result \reftagform@2.15. ∎
Corollary 2**.**
Let with . Let be an absolutely continuous function such that and is Lebesgue integrable on , then we have
[TABLE]
where , .
Proof.
The result follows from \reftagform@2.3 and \reftagform@2.15. ∎
Corollary 3**.**
Let with . Let with . Let be an absolutely continuous function such that and be a measurable function such that for all and for some real numbers , then the inequality
[TABLE]
holds.
Proof.
Substituting \reftagform@2.5 and \reftagform@2.15 in \reftagform@2.3, we get the desired result. ∎
An improvement of \reftagform@2.15 is given in the following theorem.
Corollary 4**.**
Let with . Let be two absolutely continuous functions such that , then we have
[TABLE]
where , .
Proof.
From \reftagform@2.19, we have
[TABLE]
and similarly for we have
[TABLE]
Substituting \reftagform@2.25 and \reftagform@2.26 in \reftagform@2.16 we get the desired result \reftagform@2.24. ∎
Corollary 5**.**
Let with . Let with . Let be an absolutely continuous function such that and is Lebesgue integrable on , then we have
[TABLE]
Corollary 6**.**
Let with . Let be an absolutely continuous function such that and is measurable on such that , for some real numbers and all , then the inequality
[TABLE]
holds.
Proof.
Substituting \reftagform@2.24 in \reftagform@2.14, we get the desired result. ∎
3. More inequalities
Theorem 5**.**
Let with . Let be an absolutely continuous functions and is Lebesgue integrable on .
- (1)
If , then
[TABLE] 2. (2)
If , then
[TABLE] 3. (3)
If , then
[TABLE]
, .
Proof.
It is easy to observe from Lemma 1 that
- (1)
If , then
[TABLE]
which proves \reftagform@3.1. 2. (2)
If , then
[TABLE]
for with , which proves \reftagform@3.2. 3. (3)
If , then
[TABLE]
which proves \reftagform@3.3,
and this end the proof of the theorem. ∎
Theorem 9**.**
Let with . Let be an absolutely continuous functions such that and is measurable on such that , for some real numbers and all , then the inequality
[TABLE]
holds, where , .
Proof.
From \reftagform@1.15, we have
[TABLE]
Substituting in \reftagform@2.14, we have
[TABLE]
and this proves the required inequality. ∎
Theorem 10**.**
Let with . Let be two absolutely continuous functions such that and , then the inequality then we have
[TABLE]
where , .
Proof.
Using \reftagform@1.14, we have
[TABLE]
Now,
[TABLE]
substituting in \reftagform@3.6, we get the desired result. ∎
Theorem 11**.**
Let with . Let be an absolutely continuous functions such that and is measurable on such that for some real numbers , , then the inequality
[TABLE]
holds, where , .
Proof.
From \reftagform@1.18, we have
[TABLE]
Substituting in \reftagform@2.14, we have
[TABLE]
and this proves the theorem. ∎
4. Generalizations, Remarks and Conclusion
4.1. Boggio MVT
The MVT of Pompeiu was generalized by Boggio [6] in 1947 (see also, p.,92; [28]), where he proved the following generalization of PMVT:
Theorem 6**.**
For every real valued functions and differentiable on an interval not containing [math] and for all pairs in there exists a point such that
[TABLE]
As we notice, Boggio generalization of PMVT deals with real functions defined on real intervals not containing ’. The natural question is: Is it necessarily to exclude ’ from in the Boggio generalization?.
The answer is not necessary to exclude ’ from . For example, let , . Clearly, is differentiable and for all and . The function satisfies the assumptions of Theorem 6, for instance, choose , . Applying \reftagform@4.1, for and , we get
[TABLE]
Solving for , we get . But (?)!. Which means that it’s not necessary to exclude ’ from .
To deal with more large class of functions and intervals, we need to revise Theorem 6. For more details about several and various type of MVTs and their generalization(s) we refer the reader to [28].
Let , and be a real interval such that belong to ; the interior of with . Let be the set of all real intervals for which neither nor is ever zero on , where is a real valued differentiable function. In symbols, we may write
[TABLE]
In what follows, we revise Theorem 6 and present a new independent proof.
Theorem 7**.**
Let . For every real valued differentiable functions and defined on and all distinct pairs with , there exists a point such that \reftagform@4.1 holds.
Proof.
Let be any two distinct points in with and . Define the function , given by
[TABLE]
Clearly, is continuous on , differentiable on and
[TABLE]
Applying Rolle’s theorem, there is an such that , so that
[TABLE]
equivalently we write
[TABLE]
which means that
[TABLE]
and the proof is established. ∎
Remark 5**.**
Let with . By setting , we refer to the PMVT. More generally, for , , then \reftagform@4.1 becomes
[TABLE]
For all distinct pairs .
This type of MVT was applied to obtain Ostrowski’s type inequalities in [1], [2], [10] and [27]. For comprehensive list of results regarding Ostrowski’s inequality see the recent survey [8].
4.2. Pompeiu–Chebyshev functional
Let , . Let be three integrable functions, then the Pompeiu–Chebyshev functional can be introduced such as:
[TABLE]
If we consider , then , which is the Chebyshev functional \reftagform@1.1. Also, if , , , then , which is studied in the Sections 2 and 3.
After we proposed independently, we noticed that could be deduced from more general identity of Andreiéf’s (see [22], p.243), which reads: For two continuous functions and defined on , we have the representation:
[TABLE]
Simply, substituting , , and in \reftagform@4.3, then we obtain \reftagform@4.2.
Lemma 2**.**
Let be two absolutely continuous function on the interval . Then for any , we have
[TABLE]
where with . Provided that , for .
Proof.
Since and are absolutely continuous functions on , then is absolutely continuous on and so that
[TABLE]
for any with .
Since
[TABLE]
we get the identity
[TABLE]
Taking the modulus, we have
[TABLE]
which completes the proof. ∎
Remark 6**.**
In the previous lemma, if we choose , , , . we have
[TABLE]
for with , where , and .
Remark 7**.**
By following the same approaches considered in [9] and in the Sections 2 and 3 of this work one can state more general results concerning . We left this part to the interested reader and focused researchers.
Definition 1**.**
A real valued function defined on is called increasing (decreasing) with respect to a non-negative function or simply -increasing (-decreasing) if and only if
[TABLE]
whenever , for every . In special case if we refer to the original monotonicity. Also, if we have
[TABLE]
which used in Theorem 2.
Next result generalize the inequality \reftagform@2.1 and the Chebyshev first inequality (see \reftagform@4.11):
Theorem 12**.**
Let with . Let be three integrable functions on . If is integrable on and and are both -increasing or -decreasing on , then
[TABLE]
Proof.
Assume If and are both -increasing on , then we have
[TABLE]
for all , therefore we have
[TABLE]
But also we have
[TABLE]
which proves the inequality \reftagform@4.10. ∎
Remark 8**.**
In \reftagform@4.10, if , , we recapture the first Chebyshev inequality, which reads:
[TABLE]
If , , then we recapture \reftagform@2.1.
Corollary 7**.**
Let with . Let be two integrable functions on . If is integrable on , is -increasing and is -decreasing on , then
[TABLE]
Proof.
The proof goes likewise the proof of Theorem 12. ∎
A generalization of Theorem 3, the celebrated pre-Grüss inequality is incorporated in the following theorem.
Corollary 8**.**
Let with . Let be two integrable functions on . If is integrable on , then
[TABLE]
or equivalently we can write
[TABLE]
Both inequalities are sharp.
Proof.
Since
[TABLE]
Utilizing the triangle inequality and then the Cauchy-Bunyakovsky-Schwarz (CBS) inequality, we get
[TABLE]
On the other hand, we have
[TABLE]
and similarly,
[TABLE]
Substituting in \reftagform@4.14, we get the required result. ∎
Remark 9**.**
In \reftagform@4.13, if , , we recapture the classical version of pre-Grüss inequality, which reads:
[TABLE]
If , , then we recapture \reftagform@2.3.
Remark 10**.**
The weighted version of can be presented using Andreiéf’s weighted version of \reftagform@4.3 (see [22], p.244), which reads: For arbitrary continuous functions defined on and a positive continuous function , we have
[TABLE]
For , we get the original version of Andreiéf’s identity \reftagform@4.3. Moreover, by substituting , , and in \reftagform@4.17, then we obtain the following weighted version of Pompeiu–Chebyshev functional
[TABLE]
Remark 11**.**
*In his work [9], Dragomir considered the Chebyshev functional between two absolutely continuous mappings and , , . This can be generalized in terms of -function as mentioned above (see Lemma 2 above, and Theorem 2.1 in [9]). We left this part to the interested reader.
4.3. The reverse of CBS inequality
The Pompeiu–Chebyshev functional can be very useful to bound the reverse of CBS inequality, which it has many applications in various branches of Mathematics, Physics and Statistics. In another context, we find the following related result due to Barnett and Dragomir [5]:
[TABLE]
for , , where and are assumed to be measurable on and is Hölder continuous of order with Hölderian constant , and
[TABLE]
are the usual Lebesgue norms. Clearly, this result support our consideration to generalize the Chebyshev functional as presented in the functional . Moreover, if in \reftagform@4.23, i.e., satisfy the Lipschitz condition we get
[TABLE]
It was shown in [5] that, the constant is the best possible. This can be seen by choosing and .
Fortunately, represents (exactly) the revers of CBS inequality, therefore we can state some related results, as follows:
Theorem 8**.**
Assume that and are measurable on and , are Hölder continuous of order with Hölderian constants ; respectively, then we have
[TABLE]
for , . Provided that in .
Proof.
From \reftagform@4.23, we have
[TABLE]
The same inequality holds for . Substituting \reftagform@4.3 and that one resulting from in \reftagform@4.14 we get the required result. ∎
Corollary 9**.**
Let and assume that and are measurable on and , are Hölder continuous of order with Hölderian constants ; respectively, then we have
[TABLE]
for , .
Proof.
Setting in \reftagform@4.30. ∎
Theorem 9**.**
Assume that and are measurable on and , are Hölder continuous of order with Hölderian constants ; respectively, then we have
[TABLE]
for , . Provided that neither nor equal to [math] in .
Proof.
From \reftagform@4.23, we have
[TABLE]
The same inequality holds for . Substituting \reftagform@4.3 and that one resulting from in \reftagform@4.14 we get the required result. ∎
In [5], we find another inequality for pointwise bounded functions, which reads: If there exist constants such that and for almost every (a.e.) , then
[TABLE]
A straight forward result regarding can be deduced as follows:
Theorem 10**.**
If there exist constants , such that , and for almost every (a.e.) , then
[TABLE]
Proof.
From \reftagform@4.55, we have
[TABLE]
Similarly, for we have
[TABLE]
Substituting both inequalities in the generalized pre-Grüss inequality, i.e.,
[TABLE]
which completes the proof. ∎
Remark 12**.**
-bound for can be obtained using the same approach considered in the proof of Theorem 8 which is dependent mainly on the inequality \reftagform@2.18. In this case the obtained bound will be better than that one obtained in \reftagform@4.30.
5. Some ramified inequalities
In this section we highlight the role of Pompeiu–Chebyshev functional in performing and obtaining new integral inequalities. Namely, by employing the functional some Hardy’s type inequalities are deduced. Another inequalities for differentiable functions are considered.
5.1. Hardy–Chebyshev functional
If is nonnegative -integrable function on , then is integrable over the interval for each positive and
[TABLE]
The inequality \reftagform@5.1 is known in literature as Hardy Integral inequality, which was proved by Hardy in [13]. A simple and elegant proof that is closely to Hardy original ideas and appealing to Pólya’s simplification that avoids technical details can be found in [15].
Another inequality due to Hardy [12], is that
[TABLE]
where is assumed to be nonnegative and integrable function on and .
Almost one hundred year passed from the first result of Hardy \reftagform@5.1. Through the last three decades, several applications specially in differential inequalities which play a main role in the theory of ordinary and partial differential equations have been implemented and investigated. For improvements, generalizations, extensions and useful applications of Hardy’s inequality \reftagform@5.1 the reader may refer to [14], [16], [17] and [18].
Next, we use Pompeiu–Chebyshev functional to study some inequalities of Hardy’s type on bounded real interval under some other circumstances. The approach considered here, seems to be the first work detect or figure out the application of Chebyshev functional and its generalizations in studying Hardy type inequalities.
(1.) Let be positive real numbers with . Let be an absolutely continuous on . In \reftagform@4.2, choose , with for all and . Assume is integrable on then we introduce the Hardy–Chebyshev functional
[TABLE]
or equivalently, we write .
For instance, assume the assumptions of Theorem 12 are fulfilled by our choice of and as above, then we have the *Hardy type inequality *
[TABLE]
where is Logarithmic mean. The inequality \reftagform@5.4 is sharp. Moreover, the inequality \reftagform@5.4 is reversed if we applied Corollary 7 instead of Theorem 12.
On the other hand, since , then using the bounds in \reftagform@1.9, we can state the following bounds:
Proposition 1**.**
Let . Let be an absolutely continuous function on . For , and , , we have
[TABLE]
where , is the generalized Logarithmic mean.
On utilizing the pre-Grüss inequality \reftagform@4.13 (with ), we have the following result.
Proposition 2**.**
Let . Let be an integrable on . Then
[TABLE]
where
[TABLE]
The Hardy–Chebyshev functional \reftagform@5.3 can be extended to be of Pompeiu–Chebyshev type as follows:
[TABLE]
Clearly, .
(2.) To generalize the previous presentations and the therefore to get more general inequalities of Hardy type, let and assume is non-negative. Therefore,
[TABLE]
or equivalently, we write , for .
Assume the assumptions of Theorem 12 are fulfilled by our choice of and as in the previous part (1), then we have the following new inequality of Hardy’s type on bounded intervals:
[TABLE]
where is the generalized Logarithmic mean. The inequality \reftagform@5.16 is sharp. Moreover, the inequality is reversed if we applied Corollary 7 instead of Theorem 12.
On the other hand, since , then using the bounds in \reftagform@1.9, we can state the following bounds:
Proposition 3**.**
Let . Let be a non-negative and absolutely continuous function on . For , and , , we have
[TABLE]
On utilizing the pre-Grüss inequality \reftagform@4.13 (with ), we have the following result.
Proposition 4**.**
Let . Let be an integrable on . Then
[TABLE]
for , where
[TABLE]
A generalization of Hardy–Chebyshev–Pompeiu functional \reftagform@5.14 can be presented such as:
[TABLE]
Clearly, .
(3.) Another way to deal with Hardy–Chebyshev functional by defining the functions , and given by , . Making use of the Chebyshev functional , we can write:
[TABLE]
for . In case , we have the Hardy functional:
[TABLE]
and we have the following bound:
Theorem 11**.**
Let be such that is convex on . If , then the inequality
[TABLE]
holds, for all with .
Proof.
Utilizing the triangle inequality on the right hand side of the identity
[TABLE]
and using the Hölder’s inequality, we get
[TABLE]
Since is convex on , then by Hermite-Hadamard inequality; i.e.,
[TABLE]
which follows that
[TABLE]
Now, Alomari in [4] proved that if is absolutely continuous functions whose first derivative is positive and , then for any , the inequality
[TABLE]
holds for all .
Since is convex and therefore is absolutely continuous on with , then by \reftagform@5.29 we can obtain the following two inequalities which are of great interest and not less important than the main result \reftagform@5.26 itself:
[TABLE]
and
[TABLE]
where, and . Substituting the inequalities (5.30), (5.1) and (5.28), we get the required result \reftagform@5.26. ∎
Remark 13**.**
The corresponding version of Hardy’s inequality on bounded interval , is given in the inequality \reftagform@5.30 (or \reftagform@5.1). To treat this formally, we can say that: For the absolutely continuous function such that . If (or ), then we the inequality
[TABLE]
In special case, let and , then
[TABLE]
5.2. Other inequalities
In \reftagform@4.2, choose , with , for all , and (where is assumed to be differentiable in this case), then the following identity can be ramified from \reftagform@4.2
[TABLE]
Thus several inequalities can be obtained for this functional.
For instance, assume the assumptions of Theorem 12 are fulfilled by our choice of and as above, then we have the new inequality
[TABLE]
where is the arithmetic mean and is the Logarithmic mean. The inequality is reversed if we applied Corollary 7 instead of Theorem 12.
Also, since , then we note that
[TABLE]
For instance, applying \reftagform@4.10 for \reftagform@5.34, we get
[TABLE]
The inequality is reversed if we applied Corollary 7 instead of Theorem 12.
Also, in \reftagform@4.2, choose , with for all and (where is assumed to be twice differentiable in this case), then the following identity can be ramified from \reftagform@4.2
[TABLE]
Using the results in Sections 2, 3 and 4, several inequalities can be obtained for this functional.
Among others, assume the assumptions of Theorem 12 are fulfilled by our choice of and as above, then we have the new inequality
[TABLE]
The inequality is reversed if we applied Corollary 7 instead of Theorem 12.
Hence, by choosing any function we can state various inequalities. For example, let
- (1)
, then we have
[TABLE] 2. (2)
, then we have
[TABLE] 3. (3)
, , then we have
[TABLE] 4. (4)
, then we have
[TABLE]
All inequalities are reversed if we applied Corollary 7 instead of Theorem 12.
Remark 14**.**
The Pompeiu–Chebyshev functional is very rich and fruitful to generate family of integral inequalities involving functions and their derivatives, as we have seen in the Section 5.
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