A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

Zakarya, M.; Saied, Ahmed I.; ALNemer, Ghada; El-Hamid, H. A. Abd; Rezk, H. M.

doi:https://doi.org/10.1155/2022/8720702

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 8720702 | https://doi.org/10.1155/2022/8720702

A Study on Some New Generalizations of Reversed Dynamic Inequalities of Hilbert-Type via Supermultiplicative Functions

M. Zakarya,^1,2Ahmed I. Saied,³Ghada ALNemer,⁴H. A. Abd El-Hamid,⁵and H. M. Rezk⁶

Academic Editor: Samuel Nicolay

Received16 Feb 2022

Revised15 May 2022

Accepted20 Jun 2022

Published16 Aug 2022

Abstract

In this article, we establish some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions by applying reverse Hölder inequalities with Specht’s ratio on time scales. We will generalize the inequalities by using a supermultiplicative function which the identity map represents a special case of it. Also, we use some algebraic inequalities such as the Jensen inequality and chain rule to prove the essential results in this paper. Our results (when ) are essentially new.

1. Introduction

In [1], Hardy established that where with and . In [2], Hardy showed that where and are measurable nonnegative functions such that and The constant is the best possible.

In [3], Hölder proved that where and are positive sequences and such that The integral form of (3) is where

In [4], Zhao and Cheung showed that if and are nonnegative continuous functions and is integrable on , then with where is the Specht’s ratio function (see [5]) and defined by

Also, they proved that if and , then where

In addition to this, they proved the discrete case of (8) as follows: where and

In [6], the authors proved the reverse Hilbert-type inequalities by using the Specht’s ratio as follows: if and are nonnegative and decreasing sequences of real numbers for and with , , then where where

In 1990, Ario and Muckenhoupt [7] proved an inequality which maps from to and contains one weighted function that they characterized such that the inequality holds for all nonnegative nonincreasing measurable function on with a constant independent on (here ). The characterization reduces to the condition that the function satisfies

In 1993, Sinnamon [8] generalized (14) to map from to which has two different weighted functions and characterized the weights such that the inequality holds for all measurable and the constant is independent of (here and ). The characterization reduces to the condition that the nonnegative functions and satisfy

Also, many authors study the inequalities with weighted functions (see [9–16]).

For the discrete case of (14), in 2006, Bennett and Gross-Erdmann [11] characterized the weights such that the inequality holds for all nonnegative nonincreasing sequence and The characterization reduces to the condition that the nonnegative sequence satisfies

In the last decades, a time scale theory has been discovered to unify the continuous calculus and discrete calculus. A time scale is defined as an arbitrary nonempty closed subset of the real numbers . Many authors proved some dynamic inequalities on time scales which unify the inequalities in the continuous calculus and discrete calculus (see [17–26]). In particular, in 2021, Saker et al. [25] unified the inequalities (14) and (18) to be general inequality on time scales and proved that is a time scale with and Furthermore, assume that is a nonincreasing function and Suppose there is a constant with

Then,

As special cases of (20), when (, , ), we obtain the inequality (14) and when (, , ), we have the inequality (18).

Also, we can get new inequalities, for example, when ( , ), where , we get a new inequality in a new calculus of (21) in the form

For some reversed dynamic inequalities of Hilbert-type, Hölder-type, and Hardy-type on time scale, see the papers ([27–32]).

The goal of this manuscript is to use reverse Hölder inequalities with Specht’s ratio on to develop some new generalizations of reverse Hilbert-type inequalities via supermultiplicative functions on time scales.

The following is a breakdown of the paper’s structure. In Section 2, we cover some fundamentals of time scale theory as well as several time scale lemmas that will be useful in Section 3, where we prove our findings. As specific examples (when ), our major results yield (11) proved by Zhao and Cheung [6].

2. Preliminaries and Fundamental Axioms

For completeness, we recall the following concepts related to the notion of time scales. For more details of time scale analysis, we refer the reader to the two books by Bohner and Peterson [33, 34] which summarize and organize much of the time scale calculus. A time scale is an arbitrary nonempty closed subset of the real numbers .

First, we define , introduces the set of all such rd-continuous functions, and for any function , the notation denotes

The derivative of the product and the quotient (where ) of two differentiable functions and are given by

The integration by parts formula on is

The time scale chain rule ([9], Theorem 1.87) is where it is supposed that is continuously differentiable and is differentiable.

Definition 1 (see [35]). A function is supermultiplicative if When is the identity map (i.e., ), the inequality (26) holds with equality. is said to be a submultiplicative function if the last inequality has the opposite sign.

Lemma 2. If , , and , then

Proof. Using (25) on the term , we get Since , we have (note ) that Substituting (29) into (28), we see that Integrating (30) over from to , we have That is, which is (27).

Lemma 3 (Specht’s ratio [5]). Let be positive numbers, , and
Then, where The function is strictly decreasing for and strictly increasing for In addition, the following equations are true:

Lemma 4 ([16, when ]). Let such that be integrable on . If and , then where and .

Lemma 5 (Jensen’s inequality, alnamer). Assume that and . If , , and is continuous and convex, then

The inequality (37) is reversed when is continuous and concave.

Lemma 6. Let be positive and decreasing functions be positive and nondecreasing functions, and In addition, assume that are positive, increasing, concave, and supermultiplicative functions. If with then

Proof. For , we have and then (where )

Since is decreasing, we have from (41) (where ) that

Using the facts that , is an increasing function and (42), we get and then, we obtain (where and is nondecreasing) that

Thus, the function is decreasing. Therefore, we have for that and then,

Integrating the last inequality over from to , we have and then,

Since the function is decreasing, we have for that and then,

By integrating (50) over from to , we have Thus,

From (48) and (52), we observe that

Since is decreasing on and increasing on , we get that one of is maximum (where ), and it is in the form which is (38). Similarly, with respect to when , we get which is (39).

Throughout the article, we will assume that the functions are nonnegative rd-continuous functions on and the integrals considered are assumed to exist. Now, we will present and justify our main findings.

3. Main Results

Theorem 7. Let , , and be positive and decreasing functions. Assume that are positive functions such that are increasing, concave, and supermultiplicative functions with where are positive constants. If are positive and nondecreasing functions and with , then holds for all , where with such that

Proof. Applying (27) with , we have

Multiplying the last inequality by we get (where is nondecreasing) that

From Lemma 6, the last inequality becomes

Similarly, we have for the decreasing function , the nondecreasing function and , that

From (57), (67), and (68), we get (note is a positive, increasing and supermultiplicative function) that and then, by applying the Jensen inequality on the right hand side of (69) (where is a concave function), we have that

Similarly, with respect to (57) and (68), we see (where is a positive, increasing, concave, and supermultiplicative function) that

Multiplying (70) and (71), we see that

Applying (36) on the right hand side of (72), we observe that

Multiplying (73) by and then, taking the integration over from to and the integration over from to , we get to obtain

Applying the integration by parts formula on the term with and , we get where and then where ),

Similarly, we see that

Substituting (79) and (80) into (76), we observe that and then, by applying (36), (79), and (80), we see that and thus, we have from (57) and (60) that which is (58).

Remark 8. If , then and then we have for , such that that

Remark 9. If , , and , the inequality (58) becomes This inequality is new in continuous calculus.

Remark 10. As a special case of (84) when , , , , and , we obtain (11) demonstrated in [36].

4. Conclusion and Future Work

In this paper, we studied some new generalizations of reversed dynamic inequalities of Hilbert-type via supermultiplicative functions on time scales by applying reverse Hölder inequalities and the Specht’s ratio function. We generalized the inequality proved by Zhao and Cheung [6] by using a supermultiplicative function which the identity map represents a special case of it and also by using the power greater than when we used different powers with , where we get the special case when Also, we added some increasing functions for generalizing the results where we get the results when we take the identity function as a special case. In the future, we will continue to generalize more reversed dynamic inequalities of Hilbert-type by using Kantorovich’s ratio and n-tuple fractional integral. In particular, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann-Liouville type on time scales. In addition to this, we may generalize these results to be with multidimensional Hilbert-type inequalities via supermultiplicative functions on time scales. It will also be very enjoyable to introduce such inequalities in quantum calculus.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

Software and writing-original draft were contributed by H. M. Rezk, G. AlNemer, and A I. Saied. Writing-review and editing was contributed by H. M. Rezk and M. Zakarya. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This research project was funded by the Deanship of Scientific Research, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant No. PRFA-P-42-14.

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Copyright © 2022 M. Zakarya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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