1. Introduction
In the time (1862–1943), David Hilbert proved Hilbert’s double series inequality without an exact determination of the constant in his lectures on integral equations. If
and
are two real sequences such that
and
then
In 1911, Schur [
1] proved that
in (
1) is sharp and also discovered the integral analogue of (
1), which became known as the Hilbert integral inequality in the form
where
and
are measurable functions such that,
and
.
In 1925, by introducing one pair of conjugate exponents
with
Hardy [
2] gave an extension of (
1) as follows. If
such that
and
then
where the constant
in (
3) is sharp. In 1934, Hardy et al. [
3] proved the equivalent integral analog of (
3) in the form
where
and
are measurable non-negative functions such that
and
.
In 1998, Pachpatte [
4] gave a new inequality closed to that of Hilbert as follows: Let
and
with
Then
where
and
In 2000, Pachpatte [
5] generalized (
5) by introducing one pair of conjugate exponents
, such that
with
. Then, it is established that if
and
with
then
where
and
In 2002, Kim et al. [
6] generalized (
6) and proved that if
and
with
then
where
and
Furthermore, the researchers [
6] proved the continuous analog of (
7) and proved that if
and
are real continuous functions on the intervals
respectively, and let
Then
for
, where
In recent decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. Many authors proved some dynamic inequalities of Hilbert type, its generalizations and also the reversed forms, see the papers [
7,
8,
9].
The goal of this article is to use time scales nabla calculus to prove various dynamic inequalities for Hilbert-type. Furthermore, we establish some generalized inequalities of Hilbert type by using submuliplicative and bounded functions.
The organization of the paper as follows. In
Section 2, we show some basics of the time scale theory and some lemmas needed in
Section 3 where we prove our results. Our results as special cases give the inequalities (
7) and (
8) proved by Kim et al. [
6].
2. Preliminaries
For a time scale
, we define the backward jump operator as following
Let
be a function, we say that
is ld-continuous if it is continuous at each left dense point in
and the right limit exists as a finite number for all right dense points
The set of all such ld–continuous functions is ushered by
and for any function
, the notation
denotes
For more information about the time scale calculus, see [
10,
11].
The nabla derivative of
and
(where
) are
and
Definition 1 ([
10]).
A function is called a nabla antiderivative of provided holds . We then define the integral of Ξ by Theorem 1 ([
10]).
If and , then- (1)
- (2)
- (3)
The integration by parts formula on
is
The Hölder inequality on
is
where
and
Definition 2 ([
12]).
A function is sub-multiplicative ifThe inequality (11) holds with equality when G is the identity map (i.e., ). Lemma 1 (Young’s inequality [
13]).
Let be real numbers. Then we have for and that Lemma 2. Let Then Proof. Applying Lemma 1 with
and
we obtain that
which is (
12). □
In the following, we present Jensen’s inequality in the time scale nabla calculus which is a special case of ([
14] Theorem 3.3) by taking
Lemma 3. Let and . If is ld-continuous and is continuous and convex, then 3. Main Results
Throughout the article, we will assume that the functions are ld-continuous functions on and the integrals considered are assumed to exist.
Theorem 2. Let and Assume that and are non-negative functions such that are increasing, convex and submultiplicative functions withwhere are positive constants such that Ifthen we have for thatwhere and Proof. From (
14) and (
13) and using the fact that
is a non-negative, increasing, submultiplicative and convex function, we have
Applying (
10) on the term
with indices
and
we observe that
From (
17) and (
18), we obtain
From (
19) and (
20), we observe that
From Lemma 2, the inequality (
21) becomes
Dividing the two sides of (
22) on the term
and then integrating with respect to
from
to
and for
from
to
to obtain
Applying (
10) on the R.H.S of (
23), we see that
By applying the Formula (
9), we have that
and also,
and then substituting into (
24), we obtain
From (
14) and (
16), the last inequality becomes
which is (
15). □
Corollary 1. Let and Then we have for thatfor Proof. Since
we have
and
By applying (
10) on L.H.S of (
26) and (
27), we obtain
and also,
From (
28) and (
29), we see that
Applying (
12) with
we observe that
Substituting (
31) into (
30), we obtain
By dividing (
32) on the term
and then integrating with respect to
from
to
and for
from
to
to obtain
Applying (
10) on the two parts of the right term of (
33), we see
Applying (
9) on the term
with
and
we obtain
where
and then
Similarly, by applying (
9) on the term
we observe that
Substituting (
35) and (
36) into (
34), to obtain
which is (
25). □
Remark 1. As particular cases of Corollary 1, (when , ), we obtain the inequality (7) and (when , ), we obtain the inequality (8). In what follows, we generalize Corollary 1 by using a submultiplicative function.
Corollary 2. Let with Assume that and are increasing, convex and submultiplicative functions. Iffor thenwhere and Proof. Using the fact that
is a non-negative, increasing and submultiplicative function, we have
Applying (
13) on the R.H.S of (
38), we observe that
Applying (
10) on the term
with indices
and
we obtain
From (
39) and (
40), we obtain
From (
41) and (
42), we observe that
From Lemma 2, the inequality (
43) becomes
Dividing the two sides of (
44) on the term
and then integrating with respect to
from
to
and for
from
to
to obtain
Applying (
10) on the R.H.S of (
45), we see that
By applying (
9), we have that
and also,
and then substituting into (
46), we obtain
which is (
37). □
Remark 2. As a specific case of Corollary 2, when we obtain Corollary 1.
4. Conclusions and Future Work
In this work, we explored some new generalization inequalities of Hilbert type on time scales by using nabla calculus, which are used in various problems involving symmetry. Further, we also applied our inequalities to discrete and continuous calculus to obtain some new Hilbert type inequalities as special cases. Moreover, some new inequalities as special cases are discussed. In future work, we will continue to generalize more dynamic inequalities by conformable fractional calculus on time scales by using Specht’s ratio, Kantorovich’s ratio and n-tuple fractional integral. It will also be very enjoyable to introduce such inequalities on quantum calculus.
Author Contributions
Software and Writing—original draft, G.A., R.B. and H.M.R.; Writing—review and editing, A.I.S., O.B. and M.Z. All authors have read and agreed to the published version of the manuscript.
Funding
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Schur, I. Bernerkungen sur theorie der beschrankten Bilinearformen mit unendlich vielen veranderlichen. J. Math. 1911, 140, 1–28. [Google Scholar]
- Hardy, G.H. Note on a Theorem of Hilbert Concerning Series of Positive Term. Proc. Lond. Math. Soc. 1925, 23, 45–46. [Google Scholar]
- Hardy, G.H.; Littlewood, J.E.; Pólya, G. Inequalities, 2nd ed.; Cambridge University Press: Cambridge, UK, 1934. [Google Scholar]
- Pachpatte, B.G. A note on Hilbert type inequality. Tamkang J. Math. 1998, 29, 293–298. [Google Scholar] [CrossRef]
- Pachpatte, B.G. Inequalities Similar to Certain Extensions of Hilbert’s Inequality. J. Math. Anal. Appl. 2000, 243, 217–227. [Google Scholar] [CrossRef] [Green Version]
- Kim, Y.H.; Kim, B.I. An Analogue of Hilbert’s inequality and its extensions. Bull. Korean Math. Soc. 2002, 39, 377–388. [Google Scholar] [CrossRef] [Green Version]
- Ahmed, A.M.; AlNemer, G.; Zakarya, M.; Rezk, H.M. Some dynamic inequalities of Hilbert’s type. J. Funct. Spaces 2020, 2020, 4976050. [Google Scholar] [CrossRef] [Green Version]
- AlNemer, G.; Saied, A.I.; Zakarya, M.; Abd El-Hamid, H.A.; Bazighifan, O.; Rezk, H.M. Some New Reverse Hilbert’s Inequalities on Time Scales. Symmetry 2021, 13, 2431. [Google Scholar] [CrossRef]
- O’Regan, D.; Rezk, H.M.; Saker, S.H. Some dynamic inequalities involving Hilbert and Hardy-Hilbert operators with kernels. Results Math. 2018, 73, 1–22. [Google Scholar] [CrossRef]
- Bohner, M.; Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications; Birkhäuser: Boston, MA, USA, 2001; p. 96. [Google Scholar]
- Bohner, M.; Peterson, A. Advances in Dynamic Equations on Time Scales; Birkhäuser: Boston, MA, USA, 2003. [Google Scholar]
- Sandor, J. Inequalities for multiplicative arithmetic functions. arXiv 2011, arXiv:1105.0292,. [Google Scholar]
- Young, W.H. On class of summable functions and there Fourier series. Proc. R. Soc. Lond. A 1912, 87, 225–229. [Google Scholar]
- Ammi, M.R.S.; Ferreira, R.A.; Torres, D.F. Diamond-Jensen’s inequality on time scales. J. Inequal. Appl. 2008, 2008, 576876. [Google Scholar] [CrossRef] [Green Version]
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