GENERALIZATIONS OF MINKOWSKI AND BECKENBACH–DRESHER INEQUALITIES AND FUNCTIONALS ON TIME SCALES

. We generalize integral forms of the Minkowski inequality and Beckenbach–Dresher inequality on time scales. Also, we investigate a converse of Minkowski’s inequality and several functionals arising from the Minkowski inequality and the Beckenbach–Dresher inequality.


Introduction and Preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Stefan Hilger [7] in order to unify the theory of difference equations and the theory of differential equations. For an introduction to the theory of dynamic equations on time scales, we refer to [3,8]. Martin Bohner and Gusein Sh. Guseinov [4,5] defined the multiple Riemann and multiple Lebesgue integration on time scales and compared the Lebesgue ∆-integral with the Riemann ∆-integral.
In [2] Bibi et al., obtain integral forms of Minkowski's and Beckenbach-Dresher inequality on time scales.
In this paper we generalize these inequalities and investigate functional obtained from our new inequalities.

Minkowski Inequalities
. , x k ), are real valued functions of l, m, and k variables, respectively. Let (X, M, µ ∆ ) and (Y, L, ν ∆ ) be time scale measure spaces. Then, throughout in the following sections, we use the following notations: , are defined on X, Y , and X × Y , respectively. In the sequel, we assume that all occurring integrals are finite.

Theorem 2.2 (Converse of integral Minkowski inequality). Suppose
Then by using Fubini's theorem (Theorem 1.1) and the converse Hölder inequality (Theorem 1.4) on time scales, we get

Dividing both sides by
p , we obtain (2.5). For 0 < p < 1 and p < 0, the corresponding results can be obtained similarly.
Proof. By putting p = r/s and replacing F k by F s k in (2.5), raising to the power of 1 s and dividing by we get the above result.

Minkowski Functionals
In this section, we will consider some functionals which arise from the Minkowski inequality. Similar results (but not for time scales measure spaces) can be found in [9].
Let F k and V m be fixed functions satisfying the assumptions of Theorem 2.1. Let us consider the functional where U l is a nonnegative function on X such that all occurring integrals exist. Also, if we fix the functions F k and U l , then we can consider the functional where V m is a nonnegative function on Y such that all occurring integrals exist.
(iv) Suppose V m1 and V m2 are nonnegative functions such that V m2 ≥ V m1 . If p ≥ 1, then
Proof. First we show (i). We have Using the Minkowski inequality (1.3) for integrals (Theorem 1.3) with p replaced by 1/p, we have So, M 1 is superadditive for p ≥ 1 or p < 0, and it is subadditive for 0 < p ≤ 1. The proof of (ii) is similar: After a simple calculation, we have Using the Minkowski inequality (2.2) for integrals (Theorem 2.1), we have that this is nonnegative for p ≥ 1 and nonpositive for p < 1 and p = 0. Now we show (iii). If p ≥ 1 or p < 0, then using superadditivity and and the proof of (3.1) is established. If 0 < p < 1, then using subadditivity and negativity of The proof of (iv) is similar.
Remark 3.2. Put X, Y ⊆ N, then for fixed F k and U l , the function M 2 has the form provided all occurring sums are finite.
and if 0 < p < 1, then the above inequality is reversed.
and if p < 1 and p = 0, then the above inequality is reversed.
Corollary 3.2. If V m1 and V m2 are nonnegative functions such that V m2 ≥ V m1 , then The next result gives another property of M 1 , but a similar result can also be stated for M 2 .
and if 0 < p < 1, then the above inequality is reversed.
Proof. We show this only for p ≥ 1 as the other case follows similarly. Since ϕ is concave, we have

Now, from (3.1) and (3.3), we have
and the proof is established. Let and where A ⊆ X and B ⊆ Y .
The following theorem establishes superadditivity and monotonicity of the mappings M 3 and M 4 .
and if 0 < p < 1, then the above inequality is reversed.
(ii) Suppose A 1 , A 2 ⊆ X and A 1 ⊆ A 2 . If p ≥ 1 or p < 0, then and if 0 < p < 1, then the above inequality is reversed.
and if p < 1 and p = 0, then the above inequality is reversed.
(iv) Suppose B 1 , B 2 ⊆ Y and B 1 ⊆ B 2 . If p ≥ 1, then and if p < 1 and p = 0, then the above inequality is reversed.
The proof of Theorem 3.3 is omitted as it is similar to the proof of Theorem 3.1. Then, throughout in the following sections, we use the following notations: where U l and W n are nonnegative functions on X, V m is a nonnegative function on Y , F k is a nonnegative function on X × Y with respect to the measure (µ ∆ × ν ∆ ), and G t is a nonnegative function on X × Y with respect to the measure (λ ∆ × ν ∆ ). In the sequel, we assume that all occurring integrals are finite. or s < 0, p ≤ 1 ≤ q, and p = 0, provided all occurring integrals in (4.4) exist. If 0 < s ≤ 1, p ≥ 1, q ≤ 1, and q = 0, If (4.5) holds, then the reversed inequality in (4.4) can be proved in a similar way.

Beckenbach-Dresher Functionals
Let F k , G t , U l , W n be fixed functions satisfying the assumptions of Theorem 4.1. We define the Beckenbach-Dresher functional BD(V m ) by where we suppose that all occurring integrals exist.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.