Bounds for the difference between two \v{C}eby\v{s}ev functionals
Mohammad W. Alomari
TL;DR
This paper generalizes the pre-Grüss inequality and provides new bounds for the difference between two Chebyshev functionals, enhancing understanding of their relationships in mathematical analysis.
Contribution
It introduces a generalized form of the pre-Grüss inequality and derives multiple bounds for the difference between two Chebyshev functionals.
Findings
Established a generalized pre-Grüss inequality.
Derived several bounds for Chebyshev functional differences.
Enhanced theoretical understanding of Chebyshev inequalities.
Abstract
In this work, a generalization of pre-Gr\"{u}ss inequality is established. Several bounds for the difference between two \v{C}eby\v{s}ev functional are proved.
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Taxonomy
TopicsMathematical Inequalities and Applications · Mathematics and Applications · Functional Equations Stability Results
Bounds for the difference between two Čebyšev
functionals
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 pre-Grüss inequality is established. Several bounds for the difference between two Čebyšev functional are proved.
Key words and phrases:
Čebyšev functional, Grüss inequality
2000 Mathematics Subject Classification:
26D15
1. Introduction
It is well known that for a continuous function defined on , the integral mean-value theorem (IMVT) guarantees such that
[TABLE]
On the other hand, for a monotonic function that does not change sign in the interval , the weighted IMVT reads that there exists such that
[TABLE]
If one replaces the value of in (1.2) by its value in (1.1) then we get
[TABLE]
To get weighted values in (1.3) we divide the both sides by the quantity ‘’ to get
[TABLE]
which means in such way that the weighted product of two functions equal to the product of weights of that functions.
The difference between these weights
[TABLE]
is called ‘the Čebyšev functional’, which plays an important role in Numerical Approximations and Operator Theory. For more detailed history see [17].
The most famous bounds for the Čebyšev functional are incorporated in the following theorem:
Theorem 1**.**
Let be two absolutely continuous functions, then
[TABLE]
The constants , , and are the best possible.
Many authors were studied the functional (1.5) and therefore various bounds have been implemented, for more new results and generalizations the reader may refer to [1],[2],[6],[7],[9],[12],[15] and [19].
In 2001, Cerone [10] established the following identity for the Čebyšev functional:
Theorem 1**.**
Let be such that is of bounded variation and is continuous on . Then, we have the following representation:
[TABLE]
In 2007, Dragomir [13] established three equivalent identities that generalized Cerone identity (1.14) for Riemann-Stieltjes integrals, in case of Riemann integral Dragomir representation incorporated in the following theorem.
Theorem 2**.**
Let be such that is of bounded variation and is Lebesgue integrable on . Then,
[TABLE]
The absolute difference between two integral means was studied firstly by Barnett et al. in [5] and then by Cerone and Dragomir in [8], we may summarize the obtained results, as follow:
For an absolutely continuous function defined on and for all , we have
[TABLE]
and
[TABLE]
where , and .
For a Hölder continuous function of order with constant on , we have
[TABLE]
For a function of bounded variation on , we have
[TABLE]
where, .
For recent results the reader may refer to [3], where the author used \reftagform@1.15 to obtain several bounds for the Čebyšev functional. Bounds for the difference between two Stieltjes integral means was presented in [4].
Let be any integrable function and define , such that
[TABLE]
From \reftagform@1.15, it is easy to observe the following representation of the Čebyšev functional
[TABLE]
In this work by utilizing the inequalities (1.16)–(1.24), several new bounds for the absolute Difference between two Čebyšev functional , for all are provided.
Let us start by providing the following refinements of pre-Grüss inequality, which states that for any two integrable mappings defined on , the inequality
[TABLE]
holds and sharp (see [14]). Trivially, by applying AM–GM inequality on the right hand side of \reftagform@1.32, we get
[TABLE]
We may generalize the pre-Grüss inequality \reftagform@1.32 as follows:
Theorem 3**.**
Let be two integrable mappings, then
[TABLE]
for all . The double inequality is sharp.
Proof.
Simply using the \reftagform@1.32, we have
[TABLE]
and this implies the first inequality in \reftagform@3. The second inequality follows by applying the AM–GM inequality. The sharpness follows by letting . ∎
Remark 1**.**
We note that \reftagform@3 reduces to \reftagform@1.32 by setting and , thus
[TABLE]
Consequently, the right hand of \reftagform@3 the right hand of \reftagform@1.32.
2. Bounds for bounded variation integrators
The first result regarding bounded variation integrators is presented as follows:
Theorem 4**.**
Let be such that is of bounded variation on and is absolutely continuous on , then
[TABLE]
for all , where are the usual Lebesgue norms, i.e.,
[TABLE]
and
[TABLE]
Proof.
It is known that for a continuous function on and a bounded variation on , one have the inequality
[TABLE]
Employing \reftagform@2.2 for the Cerone-Dragomir identity
[TABLE]
One has as is of bounded variation on ,
[TABLE]
In the inequality (1.16), setting , and then , , we get
[TABLE]
and
[TABLE]
Substituting (2.5) and (2.6) in (2.4), we get
[TABLE]
where we used the fact that , occurs at , therefore, . Also, we note that the last inequality holds since
[TABLE]
which proves the first inequality in (2.1).
In the inequality (1.17), replace instead of ; respectively and then instead of ; respectively, we find that
[TABLE]
and
[TABLE]
Substituting (2.8) and (2.13) in (2.4), we have respectively
[TABLE]
and similarly, we have
[TABLE]
Adding (2.27) and (2.28), we get
[TABLE]
which proves the second and the third inequalities in (2.1) ∎
Corollary 1**.**
Under the assumptions of Theorem 4, we have
[TABLE]
for all . In particular case if , we get
[TABLE]
Proof.
In Theorem 4, let and set so as we get the required result. ∎
Another result when is of -–Hölder type is as follows:
Theorem 5**.**
Let be such that is of bounded variation on and is of -–Hölder type on , for and are given. Then
[TABLE]
and
[TABLE]
for all .
Proof.
We repeat the proof of Theorem 4. So as is of bounded variation and is of -–Hölder type on , then we have
[TABLE]
which proves the first inequality. To obtain the second inequality from the above inequality we may obtain that
[TABLE]
which proves (2.50), and thus the proof is completed. ∎
Corollary 2**.**
Under the assumptions of Theorem 5, we have
[TABLE]
and
[TABLE]
for all . In particular case if , then the both inequalities (2.51) and (2.52) gives the same inequality, that is
[TABLE]
Proof.
In Theorem 5, let and set so as we get the required result. ∎
Theorem 6**.**
Let be such that is of bounded variation on and is monotonic nondecreasing on , then
[TABLE]
for all .
Proof.
As is of bounded variation on and is monotonic nondecreasing on (which implies that is absolutely continuous on ), by (2.4) we have
[TABLE]
Employing the third part of (1.24), setting and , respectively we get
[TABLE]
and
[TABLE]
Substituting (2.56) and (2.57) in (2.55), we get
[TABLE]
and thus the proof is finished. ∎
Corollary 3**.**
Under the assumptions of Theorem 6, we have
[TABLE]
for all . In particular case if , then the both inequalities (2.58) gives the same inequality, that is
[TABLE]
Proof.
In Theorem 5, let and set so as we get the required result. ∎
3. Bounds for Lipschitzian integrators
Theorem 7**.**
Let be such that is –Lipschitzian on and is an absolutely continuous on , then
[TABLE]
where, and .
Proof.
Using the fact that for a Riemann integrable function and -Lipschitzian function , one has the inequality
[TABLE]
As is –Lipschitzian on , by (3.6) we have
[TABLE]
where we used the inequality (1.16), with and ; respectively.
To obtain the second inequality, setting and ; respectively, in (1.17), we get
[TABLE]
and
[TABLE]
Substituting (3.8) and (3.9) in (3.7), we get
[TABLE]
which proves the second inequality in (3.1). ∎
Corollary 4**.**
Under the assumptions of Theorem 7, then
[TABLE]
where, and . In particular case, if then
[TABLE]
Theorem 8**.**
Let be such that is –Lipschitzian on and is of -–Hölder type on where and are given, then
[TABLE]
Proof.
We repeat the proof of Theorem 7. As is –Lipschitzian and is of -–Hölder type on , by (1.23) we have
[TABLE]
where for the last inequality a simple calculation yields that
[TABLE]
which completes the proof. ∎
Corollary 5**.**
Let be two Lipschitzian mappings on with Lipschitz constants , then
[TABLE]
Moreover,
[TABLE]
for all . In particular case if , we have
[TABLE]
Proof.
In (3.20), let we get (3.21). The inequality (3.22) can be obtained by setting , , and letting . ∎
Theorem 9**.**
Let be two absolutely continuous on . If , , , then
[TABLE]
Proof.
Taking the absolute value in (1.15) and utilizing the triangle inequality. As , by Hölder inequality we have
[TABLE]
Now, in (1.16) put and ; respectively, then
[TABLE]
and
[TABLE]
Substituting these inequalities in (3.25) we get
[TABLE]
which prove the first inequality in (3.24).
To prove the second and third inequalities in (3.24), we apply (1.17) by setting and ; respectively, then we get
[TABLE]
Similarly, we have
[TABLE]
Substituting (3.42) and (3.47) in (3.25), we get
[TABLE]
for all with and , which proves the second and the third inequalities in (3.24). ∎
Corollary 6**.**
Under the assumptions of Theorem 9, we have
[TABLE]
In particular case, if we get
[TABLE]
Remark 1**.**
For the second inequality in (3.24) we have the following particular cases:
- (1)
If and , then we have
[TABLE]
Therefore, as we have
[TABLE]
and for we have
[TABLE] 2. (2)
If and , then we have
[TABLE]
Similarly, as , we have
[TABLE]
and for we have
[TABLE]
for all with .
Remark 2**.**
In this work, all obtained bounds for the difference between two Čebyšev functional were taken under the assumption that . The same bounds hold with a few changes in the case that . Namely, replace every ‘’ (in the obtained results) by ‘’; every ‘’ (in the obtained results) by ‘’ and accordingly the differences instead of .
Remark 3**.**
All obtained bounds hold for the Čebyšev functional , this can be done by noting that as .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M.W. Alomari, Bounds for the weighted Dragomir-Fedotov functional, Moroccan J. Pure and Appl. Anal. (MJPAA), 2 (2) (2016), 65–78.
- 2[2] M.W. Alomari, New inequalities of Grüss-Lupaş type and applications to selfadjoint operators, Armen. J. Math. , 8 (1) (2016), pp. 25–37.
- 3[3] M.W. Alomari, New Čebyšev type inequalities and applications for functions of selfadjoint operators on complex Hilbert spaces, Chinese J. Math. , Volume 2014, Article ID 363050, 10 pages.
- 4[4] M.W. Alomari, Difference between two Stieltjes integral means, Kragujevac J. Math. , 38(1) (2014), 35–49.
- 5[5] N.S. Barnett, P. Cerone, S.S. Dragomir and A.M. Fink, Comparing two integral means for absolutely continuous mappings whose derivatives are in L ∞ [ a , b ] subscript 𝐿 𝑎 𝑏 L_{\infty}[a,b] and applications, Comp. and Math. Appl. , 44 (l/2) (2002), 241–251.
- 6[6] P. Cerone and S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Ineq. , 8 (1) 2014, 159–170.
- 7[7] P. Cerone and S.S. Dragomir, New bounds for the Čebyšev functional, Appl. Math. Let. , 18 2005, 603–611.
- 8[8] P. Cerone and S.S. Dragomir, Differences between means with bounds from a Riemann–Stieltjes integral, Comp. Math. Appl. , 46 (2003) 445–453