Qi Type Diamond-Alpha Integral Inequalities

: In this paper, we establish sufﬁcient conditions for Qi type diamond-alpha integral inequalities and its generalized form on time scales.


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, Yu and Qi [2] deduced the following theorem via Jensen's inequality.

Theorem 2.
If q ∈ C([α, β]), β α q(t)dt ≥ (β − α) p−1 for given p > 1, 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.
As a generalization of the differential in calculus, ∆(delta) and ∇(nabla) dynamic derivatives play a foundational role in the time scales. Recently, researchers also have provided α 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 α dynamic derivatives.
Some works in recent years established the Qi type integral inequality on time scales [5,20].
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 α = 0, 1, 1 2 , 1 3 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.

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 [α, β] T as [α, β] ∩ T. In what follows, we always suppose α, β ∈ T. We refer the readers to [9] for more details.

Definition 1.
For any s ∈ T, the forward jump operator σ : T → T is defined by σ(s) = inf{t ∈ T : t > s}, and the backward jump operator ρ : T → T is defined by As a complement, set It is obvious that σ(s) ≥ s ≥ ρ(s).

Definition 2. The graininess function
Definition 3. T k , T k is defined as follows: Property 1. If q : T → R is a continuous function.
2. If ρ(s) < s, then q is ∇-differentiable at s ∈ T k and Property 2. Suppose q 1 , q 2 are differentiable at s ∈ T k k . Then the following holds: 1. The sum q 1 +q 2 is differentiable at s and holds for any s ∈ T k . Then Q is called a delta antiderivative of q. Moreover s a q(t)∆t = Q(s) − Q(a).
If for any s ∈ T k satisfies Q ∇ (s) = q(s).
Then Q is called a nabla antiderivative of q. Moreover s a q(t)∇t = Q(s) − Q(a).
where q ∈ C rd mean q is continuous at right-dense points, and its left-sided limits exist at left-dense points.
where q ∈ C ld 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.
Finally, we list some useful properties which can be found in [9].
Property 10. If q 1 , q 2 are bounded ∆-integral over Property 11. If q 1 and q 2 are bounded functions that are ∆-integral over R with

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 .
where σ is forward jump operate.
where ρ is backward jump operate.
Following we consider G(h) 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 [β, γ] T , satisfies for all t ∈ [β, γ], where ρ is backward jump operator. Then for all s ∈ [β, γ] T k k and p ≥ 3, the following inequality holds, Proof. Set the difference

It follows from Proposition 8 and Lemma 2 that
By Lemmas 1 and 2 again, we can get Due to φ is increasing, We immediately get for all t ∈ [β, γ], where ρ is backward jump operator. Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds, Proof. Set the difference and let It follows from Proposition 8 and Lemma 2 that . By Lemmas 1 and 2 again, we can get Due to φ is increasing, We immediately get According to that G(β) = 0, we deduce G(h) ≥ 0, thereby completes the proof.
If take p 2 = p 1 − 1 in Theorem 10, we can deduce Theorem 9 immediately. By virtue Theorem 10, we obtain the following corollaries by setting α = 0 and 1.

Corollary 1. If φ is a non-negative and continuous function defined on [β, γ] T , satisfies
for all t ∈ [β, γ], where ρ is the backward jump operator. Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds,

Corollary 2.
If φ is a non-negative and continuous function defined on [β, γ] T , satisfies where ρ is backward jump operator. Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds, Theorem 11. Suppose φ is a non-negative and continuous function defined on [β, γ] T , satisfies Proof. Set the difference It follows from Proposition 8 and Lemma 2 that In this sense, Some special cases, if take α = 0 and 1, we obtain the following corollaries.

Corollary 3.
If φ is a non-negative and continuous function defined on [β, γ] T , satisfies Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds,

Corollary 4.
If φ is a non-negative and continuous function defined on [β, γ] T , satisfies Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds, Theorem 12. Suppose φ is a non-negative and continuous function defined on [β, γ] T , satisfies for all t ∈ [β, γ], where ρ is backward jump operator. Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 3, the following inequality holds, Proof. Set the difference

It follows from Proposition 8 and Lemma 2 that
. By Lemmas 1 and 2 again, we can yield According with monotony of φ, We immediately get Taking t = β in (9) we get Clearly, According to that G(β) = 0, we deduce G(h) ≥ 0, thereby completes the proof.
If we choose α = 0 or 1, we obtain the following results.

Corollary 5.
If φ is a non-negative and continuous function defined on [β, γ] T , satisfies Then for all s ∈ [β, γ] T k k and p 1 , p 2 ≥ 2, the following inequality holds, Under the basic assumptions that p 1 , p 2 ≥ 3, φ is a non-negative and continuous function defined on [β, γ] T , based on Theorems 10 and 12, we obtain the following results for all t ∈ [β, γ].
If φ is decreasing, we can obtain the similar results using the same method. However, if φ satisfying neither (11) nor (11), whether inequality (7) or the reverse of inequality (7) holds needs further research.

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.

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].

Theorem 13 (Qi type ∆ integral inequality).
If φ ∈ C rd ([α, β] T ) is a non-negative function, and for given p > 1 satisfies Then, According to the condition, Thereby we complete the proof.
In the same way, we can deduce other two inequalities.

Theorem 14 (Qi type ∇ integral inequality).
If φ ∈ C ld ([α, β] T ) is a non-negative function, and for given p > 1 satisfies  Proof. We know that q(x) = x p where p > 1 is convex on x ≥ 0. It is obvious that According to the condition, Combining the inequalities (11) and (12), we complete the proof.

Theorem 15 (Qi type diamond-α integral inequality).
If φ ∈ C([α 1 , β 1 ] T ) is a non-negative function, and for given p > 1 satisfies Then Proof. Based on that and use Lemma 5 with q(x) = x p , we have According to the condition, Combining the inequalities (13) and (14), we complete the proof.
In particularly, if we let T = R in arbitrary one among the above three theorems, then ∆ = ∇ = α = d, 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.

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: R → (m 1 , m 2 ) is a non-negative function where n have been given, n ≥ 3, and q ∈ C((m 1 , m 2 ), R) is convex, then Lemma 7. If φ: R → (m 1 , m 2 ) is a non-negative function where n have been given, n ≥ 3, and q ∈ C((m 1 , m 2 ), R) is convex, then .

Theorem 16 (Qi type ∆-integral inequalities of two variables). If
is continuous with m 1 ≥ 0, and for given p > 1 satisfies Then Proof. Since q(x) = x p is convex and using Lemma 6 with q(x) = x p and n = 2, 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
is continuous with m 1 > 0, and for given p > 1 satisfies Proof. Since q(x) = x p is convex and using Lemma 6 with q(x) = x p and n = 3, we have . (17) 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.
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
is continuous with m 1 > 0, and for given p > 1 satisfies Proof. Since φ is non-negative, then using Lemma 7 with q(x) = x p and n = 2, 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 : [α 1 , is continuous with m 1 > 0, and for given p > 1 satisfies Then Proof. Since φ is non-negative, using Lemma 7 where q(x) = x p and n = 3, for q(x) = x p is convex when x ≥ 0, we have (21) 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. Proof. In the same way, we find R φ(s 1 , s 2 , . . . , s n )∇ 1 s 1 ∇ 2 s 2 . . . ∇ n s n ≥ 0, using Lemma 7 with q(x) = x p , we have
Next three inequalities is about Qi type diamond-α integral inequalities of two, three, and n variables.

Theorem 22 (Qi type diamond-α integral inequalities of two variables). If
is continuous with m 1 > 0, and for given p > 1 satisfies Proof. We can find that In Lemma 8, take q(x) = x p , we have According to the condition, Combining the inequalities (24) and (25), we complete the proof.
If set α equal to 1 2 or 1 3 , then we have following corollaries.
is continuous with m 1 > 0, and for given p > 1 satisfies Then, is continuous with m 1 > 0, and for given p > 1 satisfies Then Theorem 23 (Qi type diamond-α integral inequalities of three variables).
is continuous with m 1 > 0, and for given p > 1 satisfies Then Proof. In the same way, the following inequality holds.
using Lemma 8 with q(x) = x p , we have (26) According to the condition, Combining the inequalities (26) and (27), we complete the proof.
In the same way, if set α equal to 1 2 or 1 3 in the theorem above, then following corollaries hold.
If we consider α = 1 2 , we obtain the following corollary. If we consider α = 1 3 , we obtain the following corollary.

Examples
In this section, we give some examples which applied the conclusions in Sections 3 and 4.
Proof. We take T = N, φ(t) = 2 t , p = 3 and α = 1 2 in (4), then it transforms into it is always true when t ≥ 1. Based on Theorem 9, we have Thereby we can arrive to inequality (30).