1. Introduction
In 2000, Qi provided an open problem following [
1], “Qi type integral inequality” for short in this paper.
Theorem 1 (Open problem).Under what conditions does the following inequality hold,for . Yu and Qi [
2] deduced the following theorem via Jensen’s inequality.
Theorem 2. If , for given , then The open problem has attracted the interest of many authors [
3,
4,
5,
6,
7]. The analytic method and employing the Jensen’s inequality are two powerful methods for the study of Qi type integral inequality. Meanwhile, studies in the past two decades have provided some promotions of the inequality.
Pogány [
3] posed the following inequality
and gave a sufficient condition for (
2) by using the Hölder inequality.
In [
4], the authors proved the following results which strengthen the Qi type integral inequality.
Theorem 3. If is non-negative and increasing, for all , then Theorem 4. If is non-negative and increasing, for given , then Since the theory of time scales was established by Hilger [
8] in 1988, it has been used widely by many branches of sciences such as finance, statistics, physics. Moreover, many researches about the theory which unifies and gives a generalization of the discrete theory and the continuous theory have been published, such as [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18].
As a generalization of the differential in calculus,
and
dynamic derivatives play a foundational role in the time scales. Recently, researchers also have provided
as a weighting between
and ∇ dynamic derivatives. It was defined as a linear combination of
and ∇ dynamic derivatives. Readers can consult [
19] to find out more basic rules of
dynamic derivatives.
Some works in recent years established the Qi type integral inequality on time scales [
5,
20].
Theorem 5 (Qi type-integral inequality).([20]) If and ϕ is a monotonic non-negative function defined on which satisfiesfor all . Then Theorem 6 (Qi type ∇-integral inequality).([20]) If and ϕ is a monotonic non-negative function defined on which satisfiesfor all . Then Theorem 7 ([5]) If , is Δ-differential and increasing function satisfies Theorem 8 ([5]) If , is ∇-differential and increasing function satisfies However, generalizing the Qi type integral inequality to the diamond-alpha integral had been a largely under explored domain which none of works has been devoted to it.
The first aim of this paper is to determine a sufficient condition for inequality (
5) via analytic method in Theorem 9.
Then we will consider the inequalities (
2) and (
3) generalized to diamond-alpha integral cases, that is, we will determine the sufficient conditions for inequalities (
7) and (
8) in Theorems 10 and 11.
Meanwhile we also consider a sufficient condition for the reverse of inequality (
7) in Theorem 12.
Last but not least, we will give concise solutions of the open Problem 1 generalized on time scales via Jensen’s inequalities. Meantime, we will consider the cases including n variables, more precisely, special cases will be considered.
In the following part of this paper, some important and fundamental properties of time scales will be given in the
Section 2. In
Section 3 we will deduce Theorems 9–12 via analysis method. A concise method will be used to prove the Qi type high dimensional integral inequalities on time scales in
Section 4.
2. Preliminaries
We introduce some definitions and algorithms of time scales in this section. Time scales is an arbitrary nonempty closed subset of the real number and we regard
as
. In what follows, we always suppose
. We refer the readers to [
9] for more details.
Definition 1. For any , the forward jump operator σ: is defined byand the backward jump operator ρ: is defined by It is obvious that .
Definition 2. The graininess function μ: is defined by Accordingly, υ: is defined by Definition 3. , is defined as follows: Property 1. If q: is a continuous function.
If
, then
q is
-differentiable at
and
If
, then
q is ∇-differentiable at
and
If
, then
Property 2. Suppose , are differentiable at . Then the following holds:
The sum
+
is differentiable at
s and
For
.
is differentiable at
s and
Definition 4. Let : . Ifholds for any . Then Q is called a of q. Moreover If for any satisfies Then Q is called a of q. Moreover Property 3. If , : are integrable on. Then
The sum
+
is integrable on
and
For any const
k,
is integrable on
and
Property 4. If : are integrable on, then
and
Next are two particularly useful formulas.
Property 5. Let .
If
, then
where
mean
q is continuous at right-dense points, and its left-sided limits exist at left-dense points.
If
, then
where
mean
f is continuous at left-dense points, and its right-sided limits exist at right-dense points.
Corollary 2.47 in [
9] shows that there exists the relationship between monotonicity and the
-differential or the ∇-differential as follows.
Property 6. If are delta derivative at , then q is increasing (decreasing) if and only if for all .
Property 7. If are nabla derivative at , then q is increasing (decreasing) if and only if for all .
The following two propositions can be found in [
19].
Property 8. If is continuous at t, , then Property 9. If is continuous at t, , then Finally, we list some useful properties which can be found in [
9].
Definition 5. If g is Δ-integrable onthen set Property 10. If are bounded Δ-integral overand , then Property 11. If and are bounded functions that are Δ-integral over R withthen Remark 1. In particular, if , then 3. Qi Type Diamond-Alpha Integral Integral and Its Generalized Form
In this section, analysis method will be used to deduce sufficient conditions for Qi type diamond-alpha integral inequalities and its generalized forms.
We need the following lemmas which give an estimation to the differential of the power of f.
Lemma 1. ([20]). Suppose g: is a increasing function, and if , thenwhere σ is forward jump operate. Lemma 2. ([20]). Suppose g: is a increasing function, and if , thenwhere ρ is backward jump operate. Following we consider as the difference between the left hand and their right hand side, and take its nabla differential. According to the Proposition 7, we can complete the proof with analysis.
Theorem 9. If ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is backward jump operator. Then for all and , the following inequality holds, Proof. Set the difference
and let
It follows from Proposition 8 and Lemma 2 that
Since
are non-negative, and
, it is sufficient to prove that
(define
as
). By Lemmas 1 and 2 again, we can get
Clearly, , so . Because , we deduce , thereby completes the proof. □
Theorem 10. If ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is backward jump operator. Then for all and , the following inequality holds, Proof. Set the difference
and let
It follows from Proposition 8 and Lemma 2 that
Since
are non-negative, and
, it is suffices to prove that
(define
as
). By Lemmas 1 and 2 again, we can get
Clearly, , so . According to that , we deduce , thereby completes the proof. □
If take in Theorem 10, we can deduce Theorem 9 immediately.
By virtue Theorem 10, we obtain the following corollaries by setting and 1.
Corollary 1. If ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is the backward jump operator. Then for all and , the following inequality holds, Corollary 2. If ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is backward jump operator. Then for all and , the following inequality holds, Theorem 11. Suppose ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is backward jump operator. Then for all and , the following inequality holds, Proof. Set the difference
where
, and let
It follows from Proposition 8 and Lemma 2 that
Since
are non-negative, and
, it is suffices to prove that
Note that
equal to
, thus
In this sense,
so
. Since
, we deduce
, thereby completes the proof. □
Some special cases, if take and 1, we obtain the following corollaries.
Corollary 3. If ϕ is a non-negative and continuous function defined on , satisfiesfor all . Then for all and , the following inequality holds, Corollary 4. If ϕ is a non-negative and continuous function defined on , satisfiesfor all . Then for all and , the following inequality holds, Theorem 12. Suppose ϕ is a non-negative and continuous function defined on , satisfiesfor all , where ρ is backward jump operator. Then for all and , the following inequality holds, Proof. It follows from Proposition 8 and Lemma 2 that
Since
are non-negative, and
, it is suffices to prove that
(define
as
). By Lemmas 1 and 2 again, we can yield
According with monotony of
,
Taking
in (
9) we get
hence we obtain
Clearly,
so
. According to that
, we deduce
, thereby completes the proof. □
If we choose or 1, we obtain the following results.
Corollary 5. If ϕ is a non-negative and continuous function defined on , satisfiesfor all . Then for all and , the following inequality holds, Corollary 6. If ϕ is a non-negative and continuous function defined on , satisfiesfor all . Then for all and , the following inequality holds, Under the basic assumptions that , is a non-negative and continuous function defined on , based on Theorems 10 and 12, we obtain the following results for all .
Remark 2. If condition (11) holds, then (7) established; if condition (11) holds, then the reverse of (7) established. If
is decreasing, we can obtain the similar results using the same method. However, if
satisfying neither (11) nor (11), whether inequality (
7) or the reverse of inequality (
7) holds needs further research.
4. Qi Type Integral Inequalities of N Variables
In
Section 3, we use differential to observe the monotonicity of
G. It is surely a useful method that can be used in Qi type integral of
n variables. However, it will produce complex conditions. In fact, once we take the differential of one variable among
n variables, in order to ensure it is greater than or equal to 0, we need a condition. Regardless of the initial conditions, it also need
n conditions.
In this section, we use Jensen’s inequalities on time scales to deduce concise condition for Qi type inequality. It’s worth to point out that Jensen’s inequalities and other related topics are still a research hot spot recent years [
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33].
4.1. Qi Type Integral Inequalities of One Variable on Time Scales
Firstly, we list three Jensen’s inequalities of one variable on time scales, all of them have been given in [
34,
35,
36].
Lemma 3. ([34]). If and q is a continuous and convex function defined on , then Lemma 4. ([35]). If and q is a continuous and convex function defined on , then Lemma 5. ([36]). Let and . If and q is a continuous and convex function defined on , then Using lemmas, we can get following three theorems.
Theorem 13 (Qi typeintegral inequality).If is a non-negative function, and for given satisfies Proof. Consider the function
defined on
. If
, we know that
q is convex. Since
is non-negative,
using Lemma 3 with
, we have
According to the condition,
Thus,
Thereby we complete the proof. □
In the same way, we can deduce other two inequalities.
Theorem 14 (Qi type ∇ integral inequality).If is a non-negative function, and for given satisfies Proof. We know that
where
is convex on
. It is obvious that
using Lemma 4 with
, we have
According to the condition,
Combining the inequalities (
11 and (
12), we complete the proof. □
Theorem 15 (Qi type diamond-integral inequality).If is a non-negative function, and for given satisfies Proof. Based on that
and use Lemma 5 with
, we have
According to the condition,
Combining the inequalities (
13) and (
14), we complete the proof. □
In particularly, if we let in arbitrary one among the above three theorems, then , it will deduce Theorem 2.
Comparing the proofs in
Section 3 and this subsection, we find that Jensen’s inequalities not only can simplify the condition, but also it can simplify the proof. Most importantly, it keeps the condition similar. This means that we can generalize it to the case of
n dimensions.
4.2. Qi Type Integral Inequalities of Several Variables on Time Scales
In the same way, we generalize Qi type integral inequalities to higher dimensions. Firstly, we write down Jensen’s inequalities of
n variables as lemmas, them can be found in [
37,
38].
Lemma 6. ([37]). If ϕ: is a non-negative function where n have been given, , and is convex, thenwhere R is . Lemma 7. If ϕ: is a non-negative function where n have been given, , and is convex, then Lemma 8. ([38]). If ϕ: is a non-negative function where n have been given, , and is convex, then Next, we deduce Qi type -integral inequalities of two, three, n variables.
Theorem 16 (Qi type-integral inequalities of two variables).If ϕ: is continuous with , and for given satisfies Proof. Since
is convex and
using Lemma 6 with
and
, we have
According to the condition,
Combining the inequalities (
15) and (
16), we complete the proof. □
Theorem 17 (Qi type-integral inequalities of three variables).If ϕ: is continuous with , and for given satisfies Proof. Since
is convex and
using Lemma 6 with
and
, we have
According to the condition,
Combining the inequalities (
17) and (
18), we complete the proof. □
In the same way, we can generalize it to n dimensions.
Theorem 18 (Qi type-integral inequalities ofvariables).If ϕ: is continuous with , and for given satisfiesThen Proof. According to Remark 1,
using Lemma 4 with
, we have
Together with the condition
we can complete the proof. □
Next three inequalities is about Qi type ∇-integral inequalities of two, three, and n variables.
Theorem 19 (Qi type ∇-integral inequalities of two variables).If ϕ: is continuous with , and for given satisfies Proof. Since
is non-negative, then
using Lemma 7 with
and
, we have
According to the condition,
Combining the inequalities (
19) and (
20), we complete the proof. □
Theorem 20 (Qi type ∇-integral inequalities of three variables).If ϕ: is continuous with , and for given satisfies Proof. Since
is non-negative,
using Lemma 7 where
and
, for
is convex when
, we have
According to the condition,
Combining the inequalities (
21) and (
22), we complete the proof. □
We can get Qi type ∇-integral inequalities of n variables in the same way.
Theorem 21 (Qi type ∇-integral inequalities ofvariables).If ϕ: is continuous with , and for given satisfies Proof. In the same way, we find
using Lemma 7 with
, we have
Using now
the relation (
23) is satisfied. □
Next three inequalities is about Qi type diamond- integral inequalities of two, three, and n variables.
Theorem 22 (Qi type diamond-integral inequalities of two variables).If ϕ: is continuous with , and for given satisfies Proof. In Lemma 8, take
, we have
According to the condition,
Combining the inequalities (
24) and (
25), we complete the proof. □
If set equal to or , then we have following corollaries.
Corollary 7. If ϕ: is continuous with , and for given satisfies Corollary 8. If ϕ: is continuous with , and for given satisfies Theorem 23 (Qi type diamond- i
ntegral inequalities of three variables). If ϕ: is continuous with , and for given satisfies Proof. In the same way, the following inequality holds.
using Lemma 8 with
, we have
According to the condition,
Combining the inequalities (
26) and (
27), we complete the proof. □
In the same way, if set equal to or in the theorem above, then following corollaries hold.
Corollary 9. If ϕ: is continuous with , and for given satisfies Corollary 10. If ϕ: is continuous with , and for given satisfies Employing Lemma 8, we can deduce
Theorem 24 (Qi type diamond-integral inequalities ofvariables).If ϕ: is a continuous function with , and for given satisfies Proof. We can find that
using Lemma 8 with
, we have
We can complete the proof. □
If we consider , we obtain the following corollary.
Corollary 11. If ϕ: is continuous with , and for given satisfies If we consider , we obtain the following corollary.
Corollary 12. If ϕ: is continuous with , and for given satisfies 5. Examples
In this section, we give some examples which applied the conclusions in
Section 3 and
Section 4.
Example 1. Consider the inequality:where . Proof. We take
and
in (
4), then it transforms into
it is always true when
. Based on Theorem 9, we have
Thereby we can arrive to inequality (
30). □
Example 2. Consider the inequality:where and . Proof. We take
in (
6), then it transforms into
noting that
for
, so (
32) holds for
. Based on Theorem 10, we have
Thereby we can arrive at inequality (
31). □
Example 3. Consider the inequality:where . Proof. Consider the condition (
28) with
and
, then it change into
Noting that
, so (
33) is hold for
. Based on Theorem 24, we get the desired inequality. □
Example 4. Consider the inequalities:
- (i)
- (ii)
where and .
Proof. Consider the condition (
28) with
and
, then it change into
1. If
, then the left hand of (
35) become
thus (
35) hold for
. Based on Theorem 24, we have the first inequality.
2. If
, then the left hand of (
35) is bigger than (
36), thus (
35) hold for
. Based on Theorem 24 or Corollary 12, we have the second inequality. □
Remark 3. We take in (34) for example, (34) will transforms into 6. Conclusions
In this paper, we greatly promoted the research of Qi type inequality on time scales theory. More completely, we generalize Qi type inequality and its two generalized forms through the Diamond-Alpha integral. The sufficient condition for the reverse of Qi type inequality is also considered. Then we generalize Qi type inequality to higher dimension via Jessen’s inequality. In this method, concise conditions are deduced. Qi type high dimension inequality has been studied in great detail and some special cases are given as corollaries. Moreover, some examples are given to show our conclusions are useful in the end.
Author Contributions
Data curation, Z.-X.M.; funding acquisition, Y.-R.Z.; investigation, Z.-X.M.; methodology, Z.-X.M., B.-H.G., Y.-H.Y. and H.-Q.Z.; resources, Z.-X.M. and Y.-R.Z.; writing—original draft, Z.-X.M.; writing—review and editing, Y.-R.Z., B.-H.G., F.-H.W., Y.-H.Y. and H.-Q.Z. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the Fundamental Research Funds for the Central Universities under Grant MS117.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
All data generated or analysed during this study are included in this published article.
Acknowledgments
The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.
Conflicts of Interest
The authors declare that they have no competing interests.
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