Journal of Function Spaces
Volume 2019 (2019), Article ID 7584836, 11 pages
https://doi.org/10.1155/2019/7584836

Research Article

Some New Dynamic Inequalities Involving Monotonic Functions on Time Scales

S. H. Saker,¹ E. Awwad,2,3 and A. Saied³

¹Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
²Department of Mathematics, College of Science and Arts, Gurayat, Jouf University, Saudi Arabia
³Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Correspondence should be addressed to S. H. Saker; shsaker@mans.edu.eg

Received 25 March 2019; Accepted 24 April 2019; Published 8 May 2019

Academic Editor: Gestur Ólafsson

Copyright © 2019 S. H. Saker 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.

Abstract.

In this paper, we prove some new dynamic inequalities involving monotonic functions on time scales The main results will be proved by employing Hölder’s inequality, integration by parts, and a chain rule on time scales. As a special case when our results contain the continuous inequalities proved by Heinig, Maligranda, Pečarić, Perić, and Persson and when , the results to the best of the authors’ knowledge are essentially new.

1. Introduction

In 1995 Heinig and Maligranda [1] proved that if , , , is decreasing on and is increasing on with , then for any

Inequality (1) is reversed when Also in [1] they proved that if is increasing on and is decreasing on with , then, for any ,

In [1] the authors generalized (1) and proved that if , and , are positive functions, then there exists a constant such that the inequality,

holds for all nonnegative decreasing function if and only if

In [1] it is also proved that inequality (3) holds for all nonnegative increasing functions and if and only if

In 1997 Pečarić et al. [2] generalized (1) and proved that if is decreasing with and is an increasing function on such that , and is a concave, nonnegative, and differentiable function such that , then

The function is said to be decreasing if implies that for Pečarić et al. [2] also proved that if is increasing with and is increasing on , such that , then

The function is said to be increasing if implies Furthermore they also considered the case when is increasing with and is decreasing on , such that , and proved that

Finally they proved that if is decreasing with and is decreasing on , such that , then

In the last decades some authors have been interested in finding some discrete results on analogues to bounds in different fields in analysis and as a result this subject becomes a topic of ongoing research. One reason for this upsurge of interest in discrete case is also due to the fact that discrete operators may even behave differently from their continuous counterparts. In this paper, we obtain the discrete inequalities as special cases of the results with a general domain called the time scale , which is an arbitrary nonempty closed subset of the real numbers . These new results on the time scale contain the classical continuous and discrete inequalities as special cases when and and can be extended to different inequalities on different time scales such as , , for .

In recent years the study of dynamic inequalities on time scales has received a lot of attention and has become a major field in pure and applied mathematics. For more details about the dynamic inequalities on time scales, we refer the reader to the books [3–5] and the papers [6–11].

The natural question now is the following: Is it possible to prove some new inequalities with monotonic continuous functions defined on a time scale and as special cases contain the above results?

Our aim in this paper is to give the answer to this question and find the relation between the weighted functions which ensure that the inequality

holds for all nonnegative decreasing function such that , , and Also, we establish some new dynamic inequalities involving monotonic functions in the form

where is a concave, nonnegative, and differentiable function such that , and is decreasing with , and is an increasing function on , such that

The paper is organized as follows. In Section 2, we present some preliminaries concerning the theory of time scales and prove the basic lemmas that will be needed in the proofs. In Section 3, we prove the main results by using Hölder’s inequality, integration by parts, and a chain rule on time scales. Our results when give inequalities (1), (2), (3), (6), (7), (8), and (9) proved by Heinig, Maligranda, Pečarić, Perić, and Persson. When , our results are essentially new.

2. Preliminaries and Basic Lemmas

In this section, 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 [5, 12] which summarize and organize much of the time scale calculus. A time scale is an arbitrary nonempty closed subset of the real numbers . The forward jump operator and the backward jump operator are defined by , and , where . A point, , is said to be left–dense if and , is right-dense if , is left–scattered if , and is right–scattered if A function is said to be right-dense continuous (rd-continuous) provided is continuous at right dense points and, at left-dense points in , left hand limits exist and are finite. The set of all such rd-continuous functions is denoted by The product and quotient rules for the derivative of the product and the quotient (where , here ) of two differentiable functions and are given by

Let be continuously differentiable and suppose that is delta differentiable. Then is delta differentiable and there exists in the real interval for , such that

Another shape of the chain rule is the formula

A special case of (14) is

We define the time scale interval by . In this paper, we will use Cauchy (delta) integral which we can define as follows. If , then the Cauchy (delta) integral of is defined by It can be shown (see [5]) that if , then the Cauchy integral exists, , and satisfies . In case , we have

The integration on discrete time scales is defined by and then, in case , we have

The integration by parts formula on time scales is given by

Also, we have, for and , that

The Hölder inequality on time scales is given by

where , , and , . Inequality (20) is reversed if or

Definition 1. A set is convex if, for all , and , we have

A function is concave if is convex and, for all , and , we have

A function is convex if is concave.

Definition 2. Assume that is a time scale, and If implies , then is decreasing. If implies , then is increasing. As a special case when we get the classical definitions.

Now, we prove the basic lemmas that will be used to prove our main results. Throughout the paper, we assume that the functions (without mentioning) are rd-continuous nonnegative and differentiable functions, locally integrable on and the integrals considered are assumed to exist and finite.

Lemma 3. Assume that is a time scale with and , . If , then

Proof. Let Then, the left hand side of (23) can be written in the form

Integrating the right hand side of (24) by parts with

we have that

where Using the facts that and , we see that

Substituting (27) into (24), we get that

Applying the Hölder inequality with indices , and on the term

we see that

Substituting (30) into (28), where is a positive function, we obtain

Since and , are positive functions, we see that

and then we have for (note that ) that

From (31) and (33), we have

and then

which is the desired inequality (23). The proof is complete.

As in the proof of Lemma 3, we can easily prove the following dual lemma.

Lemma 4. Assume that is a time scale with and , . If , then

3. Main Results

In this section, we state and prove our main results and for simplicity, we will assume that , are positive rd-continuous functions on and is a concave and differentiable function such that We begin with the time scale version of (3).

Theorem 5. Assume that is a time scale with , Furthermore assume that is nonnegative and decreasing function such that and

If there exists a constant such that

then

Proof. Integrating the term

by parts formula with and , we have that

where Using the facts that and , we obtain

and then

Substituting (38) into (43), we see (note is decreasing) that

Applying (23) with , and on the right side of (44), we see that

Using the assumption that , we get

Substituting (46) into (44), we have

and then

which is the desired inequality (39). The proof is complete.

Remark 6. Suppose that the inequality

holds for all nonnegative decreasing functions . Then it holds when

for any fixed number and becomes

This proves the necessary condition of Theorem 5.

From Theorem 5 and Remark 6, we have the following corollary.

Corollary 7. Assume that is a time scale with , If is nonnegative and decreasing function such that , and

then there exists a constant such that the inequality

holds if and only if

Remark 8. As a special case of Corollary 7 when , we get the integral inequality (3) proved by Heinig and Maligranda [1].

Remark 9. As a special case of Corollary 7 when and , we see that the inequality

holds if and only if

and , for nonnegative and decreasing sequences when

Theorem 10. Assume that is a time scale with , Furthermore assume that is a nonnegative bounded and increasing function such that and

If there exists a constant such that

then

Proof. Applying the integration by parts on the term with and , we have

where Using the facts that , is bounded, and , we get

Since is increasing, we obtain

From (58) and (62), we see (note that is increasing) that

and then

Applying (36) with , and on the term

we have

Substituting (66) into (64), we see (note that ) that

and then

which is the desired inequality (59). The proof is complete.

Theorem 11. Assume that is a time scale with , . If is decreasing on for and is increasing on , such that , then

Proof. Denote

and

Therefore, we have from (70) and (71) that

Since is - decreasing, then we have, for , that , and then we obtain (note is increasing and ) that

Substituting (71) into (73), we have

Applying the chain rule formula (13) on the term , we see that there exists , such that

From (71), we obtain (note is increasing) that

and then is increasing on and then we have, for , that

Since is concave on , then ( is decreasing on ) and, then, we observe from (77) that

Substituting (76) and (78) into (75), we get

From (74), we have that , and then we get (note is positive and is increasing) that

and thus we obtain from (79) that

From (72), we have

Substituting (81) into (82), we see that , and therefore is decreasing on Since , we see that Since , we have from (70) that

and then , and then we have from (70), by sitting , that

which is the desired inequality (69). The proof is complete.

Remark 12. As a special case of Theorem 11 when , we get the integral inequality (6) proved by Pečarić et al. [2].

Remark 13. As a special case of Theorem 11 when , , , and , we obtain the integral inequality (1) proved by Heinig and Maligranda [1].

Remark 14. As a special case of Theorem 11 when and , we see that the discrete inequality

holds for the decreasing sequence and the increasing sequence with

Theorem 15. Assume that is a time scale with , . If is increasing on , and is increasing on , such that , then

Proof. Denote

and

Therefore, we have from (87) and (88) that

Since is increasing, then we have, for , that , and then we get (note is increasing and ) that

and thus

From (88), inequality (91) becomes

Applying the chain rule formula (13) on the term , we see that there exists , such that

From (88), we obtain (note is increasing) that

then is increasing on , and then we have, for , that

Since is concave on , then ( is decreasing on ) and then we observe from (95) that

Substituting (94) and (96) into (93), we get

From (92), we have that , and then we get (note is positive and is increasing) that

thus we obtain from (97) that

From (89), we have

Substituting (99) into (100), we see that , and therefore is increasing on Since , we see that Since , we have from (87) that

then , and then we have, from (87) by sitting , that

which is the desired inequality (86). The proof is complete.

Remark 16. As a special case of Theorem 15 when and , we get the integral inequality (7) proved by Pečarić et al. [2].

Remark 17. As a special case of Theorem 15 when , , and , we see that the discrete inequality

holds for the increasing sequence and the increasing sequence with

Theorem 18. Assume that is a time scale with , If is increasing on , , and is decreasing on , such that , then

Proof. Denote

and

Therefore, we have from (105) and (106) that

Since is - increasing, then we have, for , that , and then we obtain (note is decreasing and ) that

Substituting (106) into (108), we observe that

By applying the chain rule formula (13) on the term , we see that there exists , such that

From (106), we obtain (note is decreasing) that

then is decreasing on , and then we have for that

Since is concave on , then ( is decreasing on ) and then we observe from (112) that

Substituting (111) and (113) into (110), we get

From (109), we have that , and then we get (note is positive and is decreasing) that

and thus we obtain from (114) that

and then

From (107), we have

Substituting (117) into (118), we see that , and therefore is decreasing on Since , we see that Since , we have from (105) that

then , and then we have, from (105) by sitting , that

which is the desired inequality (104). The proof is complete.

Remark 19. As a special case of Theorem 18 when and , we get the integral inequality (8) proved by Pečarić et al. [2].

Remark 20. As a special case of Theorem 18 when , , , and , we obtain the integral inequality (2) proved by Heinig and Maligranda [1].

Remark 21. As a special case of Theorem 18 when , , and , we see that the discrete inequality

holds for the increasing sequence and the decreasing sequence with

Theorem 22. Assume that is a time scale with , If is decreasing on , , and is decreasing on , such that , then

Proof. Denote

and

Therefore, we have from (123) and (124) that

Since is decreasing, then we have, for , that , and then we obtain (note is decreasing and ) that

and then

Substituting (124) into (127), we get

By applying the chain rule formula (13) on the term , we see that there exists , such that

From (124), we obtain (note is decreasing) that

then is decreasing on , and then we have, for , that

Since is concave on , then ( is decreasing on ) and then, we observe from (131) that

Substituting (130) and (132) into (129), we get

From (128), we have that , and then we get (note is positive and is decreasing) that

thus we obtain from (133) that

and then

From (125), we have

Substituting (136) into (137), we see that , and therefore is increasing on Since , we see that Since , we have from (123) that

then , and then we have, from (123) by sitting , that

and thus

which is the desired inequality (122). The proof is complete.

Remark 23. As a special case of Theorem 22 when , we get the integral inequality (9) proved by Pečarić et al. [2].

Remark 24. As a special case of Theorem 22 when and , we see that the discrete inequality

holds for the decreasing sequence and the decreasing sequence with

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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