Triple Diamond-Alpha integral and Hölder-type inequalities

In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.


Introduction
To unify and generalize discrete and continuous analysis, in 1998, Hilger [1] introduced the theory of time scales. Since then, many researchers have studied various aspects of the theory and obtained a lot of interesting results on time scales [1][2][3][4][5][6][7][8][9][10]. The first purpose of this paper is to give the definition of the triple Diamond-Alpha integral (triple diamond-α integral or triple α -integral) for functions of three variables on time scales.
Let u(x) and v(x) be continuous real-valued functions on [a, b], and let 1 p + 1 q = 1. (I) if p > 1 and if u(x) ≥ 0, v(x) ≥ 0, then the classical Hölder inequality holds (see [11] ( 1 ) (II) if 0 < p < 1 and if u(x) > 0, v(x) > 0, then the following reverse Hölder inequality (e.g., see [12]) holds: The classical Hölder and reverse Hölder inequalities play a very important role and have wide applications in different branches of modern mathematics. A large number of papers dealing with refinements, generalizations, and applications of the Hölder and reverse Hölder inequalities and their series analogues in different ares of mathematics have appeared. For example, Agahi et al. [13] gave generalizations of the Hölder and reverse Hölder inequalities for the pseudo-integral. Zhao et al. [14] found that the Hölder inequality for the pan-integral holds if the monotone measurer is subadditive. Tian [15][16][17][18] gave some new properties and refinements of the Hölder and reverse Hölder inequalities. For more detail, the reader may consult [19][20][21][22][23][24][25].
Theorem A Assume that T is a time scale, a, b ∈ T, and a < b. If p > 1 with 1 Later, in 2005, Wong et al. [26] gave the following Hölder-type inequalities via the Deltaintegral.
Theorem C Assume that T is a time scale, a, b ∈ T, and a < b. Let 1 If p < 0 or q < 0, then inequality (5) is reversed.
The second purpose of this paper is to give the time scale versions of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales. Moreover, using the obtained results, we present a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales.

Main results
For details on time scales theory, the readers may consult [1,[3][4][5][6][7][8][9] and the references therein. Now we give the definition of triple Diamond-Alpha integral for functions of three variables.
The triple Diamond-Alpha integral is defined as an iterated integral. Suppose that T is a time scale and a i , b i ∈ T with a i < b i (i = 1, 2, 3). Let f (x 1 , x 2 , x 3 ) be a real-valued function on T × T × T. Because we need the notation of partial derivatives with respect to variables x i , we denote the time scale partial derivatives of f (x 1 , We now give the definition of these partial derivatives. Fixing x 2 , x 3 ∈ T, the diamond-α derivative of a function T → R, is denoted by f 1 α . Next, fixing x 1 , x 3 ∈ T, the diamond-α derivative of a function T → R, is denoted by f 2 α . Finally, fixing x 1 , x 2 ∈ T, the diamond-α derivative of a function T → R, If a function f has a 1 α antiderivative F 1 , F 1 has a 2 α antiderivative F 2 , and F 2 has a 3 α antiderivative F 3 , that is, By this definition it is easy to obtain the following property for the triple Diamond-Alpha integral. To prove the main results, we need the following lemmas. Lemma 2.2 (Bernoulli's inequality; see [28]) If x > 0 and p > 1, then Lemma 2.3 (Young inequality; see [11]) Let a, b > 0.

Lemma 2.5 (Schlömilch's inequality for triple Diamond-Alpha integral)
Proof Without loss of generality, we may suppose that Replacing f by f r in (13), we find Thus, the proof of Lemma 2.5 is completed. Now, we give the following Hölder and reverse Hölder inequalities for triple Diamond-Alpha integral.
Proof Case (i): Let p, q > 0 with 1 p + 1 q = 1. Without loss of generality, we may suppose that From the Young inequality (9) we get Therefore, we get the desired inequality (14).
Next, we present the following generalizations of inequalities (14) and (15).
(ii) Case I. When p > 0, q < 0, and r > 0, we find from 1 p + 1 q = 1 r that Then, by inequality (14) we get Thus from inequality (19) we obtain Case II. When p < 0 and q < 0, by the same method as in Case I, we can obtain the desired inequality (18). The proof of Theorem 2.7 is completed.
We present another generalization of inequality (14).
Proof Case (I). Without loss of generality, we may suppose that Therefore, we get the desired inequality (21). Case (II). By the same method as in Case (I) and using the reversed inequality (11), we can obtain the desired result. 1, 2, 3), and that Proof Denote ξ i = λ i k (i = 1, 2, . . . , m). Then ξ 1 + ξ 2 + · · · + ξ m = 1. Write From Theorem 2.9 and Lemma 2.5 we have for k < 1. Therefore the proof of Theorem 2.10 is completed.

Corollary 2.11
Suppose that T is a time scale, a i , b i ∈ T with a i < b i (i = 1, 2, 3), and that

Application
In this section, using the obtained results, we give the following generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales. 3), and that Proof We prove only case (I). Write (x 1 , x 2 , . Without loss of generality, we may assume that (14) for p, q > 0 and 1 Thus we obtain the desired inequality (25).

Conclusions
As is well known, the Hölder inequality and its various extensions play a very important role in mathematical analysis. In this paper, based on the definition of the triple Diamond-Alpha integral for functions of three variables, we have presented the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales. Moreover, we gave some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Finally, using the obtained results, we have obtained a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales. In the future research, we will continue to explore other inequalities for the triple Diamond-Alpha integral on time scales.