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Journal of Function Spaces
Volume 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

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; ge.ude.snam@rekashs

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 , thenThe 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 , thenThe 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 thatFinally they proved that if is decreasing with and is decreasing on , such that , thenIn 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 [35] and the papers [611].

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 byLet be continuously differentiable and suppose that is delta differentiable. Then is delta differentiable and there exists in the real interval for , such thatAnother shape of the chain rule is the formulaA special case of (14) isWe 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 bywhere , , 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 formIntegrating the right hand side of (24) by parts with we have thatwhere Using the facts that and , we see thatSubstituting (27) into (24), we get thatApplying the Hölder inequality with indices , and on the term we see thatSubstituting (30) into (28), where is a positive function, we obtainSince and , are positive functions, we see that and then we have for (note that ) thatFrom (31) and (33), we have and thenwhich 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 thatthen

Proof. Integrating the termby parts formula with and , we have that where Using the facts that and , we obtain and thenSubstituting (38) into (43), we see (note is decreasing) thatApplying (23) with , and on the right side of (44), we see that Using the assumption that , we getSubstituting (46) into (44), we have and thenwhich 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 thatthen

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 obtainFrom (58) and (62), we see (note that is increasing) that and thenApplying (36) with , and on the term we haveSubstituting (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. DenoteandTherefore, we have from (70) and (71) thatSince is - decreasing, then we have, for , that , and then we obtain (note is increasing and ) thatSubstituting (71) into (73), we have Applying the chain rule formula (13) on the term , we see that there exists , such thatFrom (71), we obtain (note is increasing) thatand then is increasing on and then we have, for , thatSince is concave on , then ( is decreasing on ) and, then, we observe from (77) thatSubstituting (76) and (78) into (75), we getFrom (74), we have that , and then we get (note is positive and is increasing) that and thus we obtain from (79) thatFrom (72), we haveSubstituting (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. DenoteandTherefore, we have from (87) and (88) thatSince is increasing, then we have, for , that , and then we get (note is increasing and ) that and thusFrom (88), inequality (91) becomesApplying the chain rule formula (13) on the term , we see that there exists , such thatFrom (88), we obtain (note is increasing) thatthen is increasing on , and then we have, for , thatSince is concave on , then ( is decreasing on ) and then we observe from (95) thatSubstituting (94) and (96) into (93), we getFrom (92), we have that , and then we get (note is positive and is increasing) that thus we obtain from (97) thatFrom (89), we haveSubstituting (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. DenoteandTherefore, we have from (105) and (106) thatSince is - increasing, then we have, for , that , and then we obtain (note is decreasing and ) thatSubstituting (106) into (108), we observe thatBy applying the chain rule formula (13) on the term , we see that there exists , such thatFrom (106), we obtain (note is decreasing) thatthen is decreasing on , and then we have for thatSince is concave on , then ( is decreasing on ) and then we observe from (112) thatSubstituting (111) and (113) into (110), we getFrom (109), we have that , and then we get (note is positive and is decreasing) that and thus we obtain from (114) that and thenFrom (107), we haveSubstituting (117) into (118), we see that , and therefore is decreasing on Since