International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 319-331

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-319

GENERALIZATIONS OF MINKOWSKI AND BECKENBACH–DRESHER

INEQUALITIES AND FUNCTIONALS ON TIME SCALES

RABIA BIBI1,∗, ANEES UR RAHMAN2 AND MUHAMMAD SHAHZAD2

1Department of Mathematics, Abbottabad University of Science and Technology, Havelian, Abbottabad,

Pakistan

2Department of Mathematics, Hazara University, Mansehra, Pakistan

∗Corresponding author: emaorr@gmail.email

Abstract. 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.

1. 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.

Let n ∈ N be fixed. For each i ∈{1, . . . ,n}, let Ti denote a time scale and

Λn = T1 × . . .×Tn = {t = (t1, . . . , tn) : ti ∈ Ti, 1 ≤ i ≤ n}

Received 2019-08-18; accepted 2019-09-24; published 2020-03-02.

2010 Mathematics Subject Classification. Primary 26D15; Secondary 26A51, 34N05.

Key words and phrases. Minkowski inequality, Beckenbach–Dresher Inequality, time scales integrals.

c©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

319

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-319


Int. J. Anal. Appl. 18 (2) (2020) 320

an n-dimensional time scale. Let µ∆ be the σ-additive Lebesgue ∆-measure on Λ
n and F be the family

of ∆-measurable subsets of Λn. Let E ∈ F and (E,F,µ∆) be a time scale measure space. Then for a

∆-measurable function f : E → R, the corresponding ∆-integral of f over E will be denoted according

to [5, (3.18)] by∫
E

f(t1, . . . , tn)∆1t1 . . . ∆ntn,

∫
E

f(t)∆t,

∫
E

fdµ∆, or

∫
E

f(t)dµ∆(t).

By [5, Section 3], all theorems of the general Lebesgue integration theory, including the Lebesgue dominated

convergence theorem, hold also for Lebesgue ∆-integrals on Λn. Here we state Fubini’s theorem for integrals

on time scales. It is used in the proofs of our main results.

Theorem 1.1 (Fubini’s theorem). Let (X,M,µ∆) and (Y,L,ν∆) be two finite-dimensional time scale mea-

sure spaces. If f : X ×Y → R is a ∆-integrable function and if we define the functions

ϕ(y) =

∫
X

f(x,y)dµ∆(x) for a.e. y ∈ Y

and

ψ(x) =

∫
Y

f(x,y)dν∆(y) for a.e. x ∈ X,

then ϕ is ∆-integrable on Y and ψ is ∆-integrable on X and∫
X

dµ∆(x)

∫
Y

f(x,y)dν∆(y) =

∫
Y

dν∆(y)

∫
X

f(x,y)dµ∆(x). (1.1)

Hölder’s inequality and Minkowski’s inequality and their converses for multiple integration on time scales

were investigated in [1]. These inequalities hold for both Riemann integrals and Lebesgue integrals on time

scales. For completeness, let us recall these inequalities from [1].

Theorem 1.2 (Hölder’s inequality [1, Theorem 6.2]). For p 6= 1, define q = p/(p − 1). Let (E,F,µ∆)

be a time scale measure space. Assume w, f, g are nonnegative functions such that wfp, wgq, wfg are

∆-integrable on E. If p > 1, then∫
E

w(t)f(t)g(t)dµ∆(t) ≤
(∫

E

w(t)fp(t)dµ∆(t)

)1/p
×
(∫

E

w(t)gq(t)dµ∆(t)

)1/q
. (1.2)

If 0 < p < 1 and
∫
E
wgqdµ∆ > 0, or if p < 0 and

∫
E
wfpdµ∆ > 0, then (1.2) is reversed.

Theorem 1.3 (Minkowski’s inequality [1, Theorem 7.2]). Let (E,F,µ∆) be a time scale measure space. For

p ∈ R, assume w, f, g, are nonnegative functions such that wfp, wgp, w(f + g)p are ∆-integrable on E. If

p ≥ 1, then

(∫
E

w(t)(f(t) + g(t))pdµ∆(t)

)1
p

≤
(∫

E

w(t)fp(t)dµ∆(t)

)1/p
+

(∫
E

w(t)gp(t)dµ∆(t)

)1/p
. (1.3)

If 0 < p < 1 or p < 0, then (1.3) is reversed provided each of the two terms on the right-hand side is positive.



Int. J. Anal. Appl. 18 (2) (2020) 321

Theorem 1.4 (Converse of Hölder’s inequality [1, Theorem 11.3]). For p 6= 1, define q = p/(p − 1). Let

(E,F,µ∆) be a time scale measure space. Assume w, f, g are nonnegative functions such that wfp, wgq,

wfg are ∆-integrable on E. Suppose

0 < m ≤ f(t)g−q/p(t) ≤ M for all t ∈ E.

If p > 1, then

∫
E

w(t)f(t)g(t)dµ∆(t) ≥ K(p,m,M)
(∫

E

w(t)fp(t)dµ∆(t)

)1/p
×
(∫

E

w(t)gq(t)dµ∆(t)

)1/q
, (1.4)

where

K(p,m,M) = |p|1/p|q|1/q
(M −m)1/p|mMp −Mmp|1/q

|Mp −mp|
. (1.5)

If 0 < p < 1 or p < 0, then (1.4) is reversed provided either
∫
E
wgqdµ∆ > 0 or

∫
E
wfpdµ∆ > 0.

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.

2. Minkowski Inequalities

Let Ul(x1,x2, . . . ,xl), Vm(x1,x2, . . . ,xm), Gk(x1,x2, . . . ,xk), 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:

Ul = Ul(x) = Ul(u1(x),u2(x), . . . ,ul(x)), (2.1)

Vm = Vm(y) = Vm(v1(y),v2(y), . . . ,vm(y)),

Fk = Fk(x,y) = Fk(f1(x,y),f2(x,y), . . . ,fk(x,y)),

where {ui(x)}li=1, {vi(y)}
m
i=1, {fi(x,y)}

k
i=1, are defined on X, Y , and X ×Y , respectively. In the sequel, we

assume that all occurring integrals are finite.

Theorem 2.1 (Integral Minkowski inequality). If p ≥ 1, then

[∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

≤
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y) (2.2)

holds provided all integrals in (2.2) exists. If 0 < p < 1 and∫
X

(∫
Y

FkVmdν∆

)p
Uldµ∆ > 0,

∫
Y

FkVmdν∆ > 0 (2.3)

holds, then (2.2) is reversed. If p < 0 and (2.3) and∫
X

F
p
kUldµ∆ > 0, (2.4)

hold, then (2.2) is reversed as well.



Int. J. Anal. Appl. 18 (2) (2020) 322

Proof. Let p ≥ 1. Put

H(x) =

∫
Y

Fk(x,y)Vm(y)dν∆(y).

Now, by using Fubini’s theorem (Theorem 1.1) and Hölder’s inequality (Theorem 1.2) on time scales, we

have

∫
X

Hp(x)Ul(x)dµ∆(x) =

∫
X

H(x)Hp−1(x)Ul(x)dµ∆(x)

=

∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)
Hp−1(x)Ul(x)dµ∆(x)

=

∫
Y

(∫
X

Fk(x,y)H
p−1(x)Ul(x)dµ∆(x)

)
Vm(y)dν∆(y)

≤
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p
(∫

X

Hp(x)Ul(x)dµ∆(x)

)p−1
p

Vm(y)dν∆(y)

=

∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

(∫
X

Hp(x)Ul(x)dµ∆(x)

)p−1
p

and hence (∫
X

Hp(x)Ul(x)dµ∆(x)

)1
p

≤
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y).

For p < 0 and 0 < p < 1, the corresponding results can be obtained similarly. �

Theorem 2.2 (Converse of integral Minkowski inequality). Suppose

0 < m ≤
Fk(x,y)∫

Y
Fk(x,y)Vm(y)dν∆(y)

≤ M for all x ∈ X, y ∈ Y.

If p ≥ 1, then

[∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

≥ K(p,m,M)
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y) (2.5)

provided all integrals in (2.5) exist, where K(p,m,M) is defined by (1.5). If 0 < p < 1 and (2.3) holds, then

(2.5) is reversed. If p < 0 and (2.3) and (2.4) hold, then (2.5) is reversed as well.

Proof. Let p ≥ 1. Put

H(x) =

∫
Y

Fk(x,y)Vm(y)dν∆(y).



Int. J. Anal. Appl. 18 (2) (2020) 323

Then by using Fubini’s theorem (Theorem 1.1) and the converse Hölder inequality (Theorem 1.4) on time

scales, we get ∫
X

Hp(x)Ul(x)dµ∆(x) =

∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)
Hp−1(x)Ul(x)dµ∆(x)

=

∫
Y

(∫
X

Fk(x,y)H
p−1(x)Ul(x)dµ∆(x)

)
Vm(y)dν∆(y)

≥K(p,m,M)
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1/p
×
(∫

X

Hp(x)Ul(x)dµ∆(x)

)p−1
p

Vm(y)dν∆(y).

Dividing both sides by
(∫
X
Hp(x)Ul(x)dµ∆(x)

)p−1
p , we obtain (2.5). For 0 < p < 1 and p < 0, the

corresponding results can be obtained similarly. �

Now we define the rth power mean M[r](Fk,µ∆) of the function Fk with respect to the measure µ∆ by

M[r](Fk,µ∆) =



(∫

X
Frk (x,y)Ul(x)dµ∆(x)∫

X
Ul(x)dµ∆(x)

)1
r

if r 6= 0,

exp
(∫

X
log Fk(x,y)Ul(x)dµ∆(x)∫

X
Ul(x)dµ∆(x)

)
if r = 0,

(2.6)

where
∫
X
Uldµ∆ > 0.

Corollary 2.1. Let 0 < s ≤ r. Then

M[r](M[s](Fk, dν∆), dµ∆) ≥ K
(r
s
,m,M

)
M[s](M[r](Fk, dµ∆), dν∆).

Proof. By putting p = r/s and replacing Fk by F
s
k in (2.5), raising to the power of

1
s

and dividing by

(∫
X

Ul(x)dµ∆(x)

)1
r
(∫

Y

Vm(y)dν∆(y)

)1
s

,

we get the above result. �

3. 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 Fk and Vm be fixed functions satisfying the assumptions of Theorem 2.1. Let us consider the functional

M1 defined by

M1(Ul) =

[∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p
−
∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x),



Int. J. Anal. Appl. 18 (2) (2020) 324

where Ul is a nonnegative function on X such that all occurring integrals exist. Also, if we fix the functions

Fk and Ul, then we can consider the functional

M2(Vm) =

∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

−
[∫

X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

,

where Vm is a nonnegative function on Y such that all occurring integrals exist.

Remark 3.1. (i) It is obvious that M1 and M2 are positive homogeneous, i.e., M1(aUl) = aM1(Ul),

and M2(aVm) = aM2(Vm), for any a > 0.

(ii) If p ≥ 1 or p < 0, then M1(Ul) ≥ 0, and if 0 < p < 1, then M1(Ul) ≤ 0.

(iii) If p ≥ 1, then M2(Vm) ≥ 0, and if p < 1 and p 6= 0, then M2(Vm) ≤ 0.

Theorem 3.1. (i) If p ≥ 1 or p < 0, then M1 is superadditive. If 0 < p < 1, then M1 is subadditive.

(ii) If p ≥ 1, then M2 is superadditive. If p < 1 and p 6= 0, then M2 is subadditive.

(iii) Suppose Ul1 and Ul2 are nonnegative functions such that Ul2 ≥ Ul1. If p ≥ 1 or p < 0, then

0 ≤ M1(Ul1) ≤ M1(Ul2), (3.1)

and if 0 < p < 1, then (3.1) is reversed.

(iv) Suppose Vm1 and Vm2 are nonnegative functions such that Vm2 ≥ Vm1. If p ≥ 1, then

0 ≤ M2(Vm1) ≤ M2(Vm2), (3.2)

and if p < 1 and p 6= 0, then (3.2) is reversed.

Proof. First we show (i). We have

M1(Ul1 + Ul2) −M1(Ul1) −M1(Ul2)

=

[∫
Y

(∫
X

fp(x,y)(Ul1 + Ul2)(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p

−
∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
(Ul1 + Ul2)(x)dµ∆(x)

−

[∫
Y

(∫
X

F
p
k (x,y)Ul1(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p

+

∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul1(x)dµ∆(x)

−

[∫
Y

(∫
X

F
p
k (x,y)Ul2(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p



Int. J. Anal. Appl. 18 (2) (2020) 325

+

∫
X

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul2(x)dµ∆(x)

=

[∫
Y

(∫
X

F
p
k (x,y)(Ul1 + Ul2)(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p

−

[∫
Y

(∫
X

F
p
k (x,y)Ul1(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p

−

[∫
Y

(∫
X

F
p
k (x,y)Ul2(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p
.

Using the Minkowski inequality (1.3) for integrals (Theorem 1.3) with p replaced by 1/p, we have

M1(Ul1 + Ul2) −M1(Ul1) −M1(Ul2)


 ≥ 0 if p ≥ 1 or p < 0,≤ 0 if 0 < p ≤ 1. (3.3)

So, M1 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

M2(Vm1 + Vm2) −M2(Vm1) −M2(Vm2)

=

[∫
X

(∫
Y

Fk(x,y)Vm1(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

+

[∫
X

(∫
Y

Fk(x,y)Vm2(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

−
[∫

X

(∫
Y

Fk(x,y)(Vm1 + Vm2)(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

.

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 6= 0. Now we show (iii). If p ≥ 1 or p < 0, then using superadditivity and

positivity of M1, Ul2 ≥ Ul1 implies

M1(Ul2) = M1(Ul1 + (Ul2 −Ul1)) ≥ M1(Ul1) + M1(Ul2 −Ul1) ≥ M1(Ul1),

and the proof of (3.1) is established. If 0 < p < 1, then using subadditivity and negativity of M1, Ul2 ≥ Ul1

implies

M1(Ul2) ≤ M1(Ul1) + M1(Ul2 −Ul1) ≤ M1(Ul1).

The proof of (iv) is similar. �

Remark 3.2. Put X,Y ⊆ N, then for fixed Fk and Ul, the function M2 has the form

M2(Vm1) =
∑
j∈Y

Vm1(j)

(∑
i∈X

Ul(i)Fk(i,j)
p

)1/p
−


∑
i∈X

Ul(i)


∑
j∈Y

Vm1(j)Fk(i,j)


p

1/p ,



Int. J. Anal. Appl. 18 (2) (2020) 326

where f(i,j) = Fk(i,j) ≥ 0. If p ≥ 1, then the mapping M2 is superadditive, and Vm2(j) ≥ Vm1(j) for all

j ∈ Y implies

0 ≤
∑
j∈Y

Vm1(j)

(∑
i∈X

Ul(i)Fk(i,j)
p

)1/p
−


∑
i∈X

Ul(i)


∑
j∈Y

Vm1(j)Fk(i,j)


p

1/p

≤
∑
j∈Y

Vm2(j)

(∑
i∈X

Ul(i)Fk(i,j)
p

)1/p
−


∑
i∈X

Ul(i)


∑
j∈Y

Vm2(j)Fk(i,j)


p

1/p

provided all occurring sums are finite.

Corollary 3.1. (i) Suppose Ul1 and Ul2 are nonnegative functions such that CUl2 ≥ Ul1 ≥ cUl2, where

c,C ≥ 0. If p ≥ 1 or p < 0, then

cM1(Ul2) ≤ M1(Ul1) ≤ CM1(Ul2),

and if 0 < p < 1, then the above inequality is reversed.

(ii) Suppose Vm1 and Vm2 are nonnegative functions such that CVm2 ≥ Vm1 ≥ cVm2, where c,C ≥ 0. If

p ≥ 1, then

cM2(Vm2) ≤ M2(Vm1) ≤ CM2(Vm2),

and if p < 1 and p 6= 0, then the above inequality is reversed.

Corollary 3.2. If Vm1 and Vm2 are nonnegative functions such that Vm2 ≥ Vm1, then

M[0]
(∫

Y

Fk(x,y)Vm1(y)dν∆(y),µ∆

)
−
∫
Y

M[0](Fk,µ∆)Vm1(y)dν∆(y)

≤ M[0]
(∫

Y

Fk(x,y)Vm2(y)dν∆(y),µ∆

)
−
∫
Y

M[0](Fk,µ∆)Vm2(y)dν∆(y), (3.4)

where M[0](Fk,µ∆) is defined in (2.6).

The next result gives another property of M1, but a similar result can also be stated for M2.

Theorem 3.2. Let ϕ : [0,∞) → [0,∞) be a concave function. Suppose Ul1 and Ul2 are nonnegative functions

such that

ϕ◦Ul1, ϕ◦Ul2, ϕ◦ (αUl1 + (1 −α)Ul2)

are ∆-integrable for α ∈ [0, 1]. If p ≥ 1, then

M1(ϕ◦ (αUl1 + (1 −α)Ul2)) ≥ αM1(ϕ◦Ul1) + (1 −α)M1(ϕ◦Ul2),

and if 0 < p < 1, then the above inequality is reversed.



Int. J. Anal. Appl. 18 (2) (2020) 327

Proof. We show this only for p ≥ 1 as the other case follows similarly. Since ϕ is concave, we have

ϕ(αUl1 + (1 −α)Ul2)) ≥ αϕ(Ul1) + (1 −α)ϕ(Ul2).

Now, from (3.1) and (3.3), we have

M1(ϕ◦ (αUl1 + (1 −α)Ul2)) ≥ M1(α(ϕ◦Ul1) + (1 −α)(ϕ◦Ul2))

≥ M1(α(ϕ◦Ul1)) + M1((1 −α)(ϕ◦Ul2))

≥ αM1(ϕ◦Ul1) + (1 −α)M1(ϕ◦Ul2),

and the proof is established. �

Let Fk, Ul and Vm be fixed functions satisfying the assumptions of Theorem 2.1. Let us define functionals

M3 and M4 by

M3(A) =

[∫
Y

(∫
A

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)

]p
−
∫
A

(∫
Y

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

and

M4(B) =

∫
B

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)1
p

Vm(y)dν∆(y)−
[∫

X

(∫
B

Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]1
p

,

where A ⊆ X and B ⊆ Y .

The following theorem establishes superadditivity and monotonicity of the mappings M3 and M4.

Theorem 3.3. (i) Suppose A1,A2 ⊆ X and A1 ∩A2 = ∅. If p ≥ 1 or p < 0, then

M3(A1 ∪A2) ≥ M3(A1) + M3(A2),

and if 0 < p < 1, then the above inequality is reversed.

(ii) Suppose A1,A2 ⊆ X and A1 ⊆ A2. If p ≥ 1 or p < 0, then

M3(A1) ≤ M3(A2),

and if 0 < p < 1, then the above inequality is reversed.

(iii) Suppose B1,B2 ⊆ Y and B1 ∩B2 = ∅. If p ≥ 1, then

M4(B1 ∪B2) ≥ M4(B1) + M4(B2),

and if p < 1 and p 6= 0, then the above inequality is reversed.

(iv) Suppose B1,B2 ⊆ Y and B1 ⊆ B2. If p ≥ 1, then

M4(B1) ≤ M4(B2),

and if p < 1 and p 6= 0, then the above inequality is reversed.



Int. J. Anal. Appl. 18 (2) (2020) 328

The proof of Theorem 3.3 is omitted as it is similar to the proof of Theorem 3.1.

Remark 3.3. For p ≥ 1, if Sm is a subset of Y with m elements and if Sm ⊇ Sm−1 ⊇ . . . ⊇ S2, then we

have

M4(Sm) ≥ M4(Sm−1) ≥ . . . ≥ M4(S2) ≥ 0

and M4(Sm) ≥ max{M4(S2) : S2 is any subset of Sm with 2 elements}.

4. Beckenbach–Dresher Inequalities

Let Ul, Vm, Fk be defined as in (4.1). Let Fn(x1,x2, . . . ,xn), Gt(x1,x2, . . . ,xt) are real valued functions

of n, and t variables, respectively. Let (X,M,µ∆), (X,M,λ∆) and (Y,L,ν∆) be time scale measure spaces.

Then, throughout in the following sections, we use the following notations:

Wn = Wn(x) = Wn(w1(x),w2(x), . . . ,wn(x)), (4.1)

Gt = Gt(x,y) = Gt(g1(x,y),g2(x,y), . . . ,gt(x,y)),

where Ul and Wn are nonnegative functions on X, Vm is a nonnegative function on Y , Fk is a nonnegative

function on X ×Y with respect to the measure (µ∆ ×ν∆), and Gt 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.

Theorem 4.1. If

s ≥ 1, q ≤ 1 ≤ p, and q 6= 0 (4.2)

or

s < 0, p ≤ 1 ≤ q, and p 6= 0, (4.3)

then

[∫
X

(∫
Y
Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q
≤
∫
Y

(∫
X
F
p
k (x,y)Ul(x)dµ∆(x)

)s
p(∫

X
Gqt (x,y)Wn(x)dλ∆(x)

)s−1
q

Vm(y)dν∆(y) (4.4)

provided all occurring integrals in (4.4) exist. If

0 < s ≤ 1, p ≥ 1, q ≤ 1, and q 6= 0, (4.5)

then (4.4) is reversed.



Int. J. Anal. Appl. 18 (2) (2020) 329

Proof. Assume (4.2) or (4.3). By using the integral Minkowski inequality (2.2) and Hölder’s inequality (1.2),

we have [∫
X

(∫
Y
Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q
≤

[∫
Y

(∫
X
F
p
k (x,y)Ul(x)dµ∆(x)

)1
p Vm(y)dν∆(y)

]s
[∫
Y

(∫
X
Gqt (x,y)Wn(x)dλ∆(x)

)1
q Vm(y)dν∆(y)

]s−1
=


∫

Y

((∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)s
p

)1
s

Vm(y)dν∆(y)


s

×


∫

Y

((∫
X

Gqt (x,y)Wn(x)dλ∆(x)
)1−s

q

) 1
1−s

Vm(y)dν∆(y)




1−s

≤
∫
Y

(∫
X

F
p
k (x,y)Ul(x)dµ∆(x)

)s
p
(∫

X

Gqt (x,y)Wn(x)dλ∆(x)
)1−s

q

Vm(y)dν∆(y).

If (4.5) holds, then the reversed inequality in (4.4) can be proved in a similar way. �

5. Beckenbach–Dresher Functionals

Let Fk, Gt, Ul, Wn be fixed functions satisfying the assumptions of Theorem 4.1. We define the

Beckenbach–Dresher functional BD(Vm) by

BD(Vm) =

∫
Y

(∫
X
F
p
k (x,y)Ul(x)dµ∆(x)

)s
p(∫

X
Gqt (x,y)Wn(x)dλ∆(x)

)s−1
q

Vm(y)dν∆(y)

−
[∫
X

(∫
Y
Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q ,
where we suppose that all occurring integrals exist.

Theorem 5.1. If (4.2) or (4.3) holds, then

BD(Vm1 + Vm2) ≥ BD(Vm1) + BD(Vm2). (5.1)

If Vm2 ≥ Vm1, then

BD(Vm1) ≤ BD(Vm2). (5.2)

If C,c ≥ 0 and CVm2 ≥ Vm1 ≥ cVm2, then

CBD(Vm2) ≥ BD(Vm1) ≥ cBD(Vm1). (5.3)

If (4.5) holds, then (5.1), (5.2) and (5.3) are reversed.



Int. J. Anal. Appl. 18 (2) (2020) 330

Proof. Assume (4.2) or (4.3). Then we have

BD(Vm1 + Vm2) −BD(Vm1) −BD(Vm2)

=

[∫
X

(∫
Y
Fk(x,y)Vm1(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm1(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q
+

[∫
X

(∫
Y
Fk(x,y)Vm2(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm2(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q
−
[∫
X

(∫
Y
Fk(x,y)Vm1(y)dν∆(y) +

∫
Y
Fk(x,y)Vm2(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
Y
Gt(x,y)Vm1(y)dν∆(y) +

∫
Y
Gt(x,y)Vm2(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q
≥ 0,

where in the last inequality we used (4.4) from Theorem 4.1. Using Theorem 4.1 again, Vm2 ≥ Vm1 implies

BD(Vm2) = BD(Vm1 + (Vm2 −Vm1)) ≥ BD(Vm1) + BD(Vm2 −Vm1) ≥ BD(Vm1).

The proof of (5.3) is similar. If (4.5) holds, then the reversed inequalities of (5.1), (5.2) and (5.3) can be

proved in a similar way. �

Let Fk, Gt, Ul, Vm, Wn be fixed functions. We define a functional BD1 by

BD1(A) =

∫
A

(∫
X
F
p
k (x,y)Ul(x)dµ∆(x)

)s
p(∫

X
Gqt (x,y)Wn(x)dλ∆(x)

)s−1
q

Vm(y)dν∆(y)

−
[∫
X

(∫
A
Fk(x,y)Vm(y)dν∆(y)

)p
Ul(x)dµ∆(x)

]s
p[∫

X

(∫
A
Gt(x,y)Vm(y)dν∆(y)

)q Wn(x)dλ∆(x)]s−1q ,
where A ⊆ Y .

For BD1, the following result holds.

Theorem 5.2. (i) Suppose A1,A2 ⊆ Y and A1 ∩A2 = ∅. If (4.2) or (4.3) holds, then

BD1(A1 ∪A2) ≥ BD1(A1) + BD1(A2),

and if (4.5) holds, then the above inequality is reversed.

(ii) Suppose A1,A2 ⊆ Y and A1 ⊆ A2. If (4.2) or (4.3) holds, then

BD1(A1) ≤ BD1(A2),

and if (4.5) holds, then the above inequality is reversed.

The proof of Theorem 5.2 is omitted as it is similar to the proof of Theorem 5.1.



Int. J. Anal. Appl. 18 (2) (2020) 331

Remark 5.1. If Sk ⊆ X has k elements and if Sm ⊇ Sm−1 ⊇ . . . ⊇ S2, then (4.2) or (4.3) implies

BD1(Sm) ≥ BD1(Sm−1) ≥ ···≥ BD1(S2) ≥ 0

and BD1(Sm) ≥ max{BD1(S2) : S2 is any subset of Sm with 2 elements}, while (4.5) implies the reversed

inequalities with max replaced by min.

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

of this paper.

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	1. Introduction and Preliminaries
	2. Minkowski Inequalities
	3. Minkowski Functionals
	4. Beckenbach–Dresher Inequalities
	5. Beckenbach–Dresher Functionals
	References