Hindawi
Journal of Mathematics
Volume 2022, Article ID 7668860, 17 pages
https://doi.org/10.1155/2022/7668860
Research Article
On Refinements of Multidimensional Inequalities of
Hardy-Type via Superquadratic and Subquadratic Functions
M. Zakarya ,1,2 Ghada AlNemer ,3 H. A. Abd El-Hamid,4 Roqia Butush,5
and H. M. Rezk 6
1
King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, 61413 Abha, Saudi Arabia
Department of Mathematics, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt
3
Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428,
Riyadh 11671, Saudi Arabia
4
Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
5
Department of Mathematics, University of Jordan, P.O. Box 11941, Amman, Jordan
6
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt
2
Correspondence should be addressed to Ghada AlNemer; gnnemer@pnu.edu.sa
Received 29 July 2022; Revised 15 September 2022; Accepted 10 October 2022; Published 24 November 2022
Academic Editor: Ding-Xuan Zhou
Copyright © 2022 M. Zakarya et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy-type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refned
Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. Te performance of the superquadratic
functions is reliable and efective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire
numerous examples and implementations of the related inequalities.
∞
∞
1 s
ds
ds
􏽚 Φ􏼒 􏽚 f(t)dt􏼓 ≤ 􏽚 Φ(f(s)) ,
s 0
s
s
0
0
1. Introduction
In [1], Hardy proved that if λ > 1, f ≥ 0 over interval (0, ∞)
∞
and 􏽒0 fλ (s)ds < ∞; then,
λ
λ
∞ 1 s
∞
λ
λ
􏽚 􏼒 􏽚 f(t)dt􏼓 ds ≤ 􏼠
􏼡 􏽚 f (s)ds,
s 0
λ− 1
0
0
(1)
where the constant (λ/(λ − 1))λ is sharp. By rewriting (1)
with f1/λ rather than f and taking the limit as λ ⟶ ∞, we
acquire the limiting case of Hardy’s inequality known as the
inequality of Pólya-Knopp ([2]), that is,
∞
∞
1 s
􏽚 exp􏼒 􏽚 ln f(t)dt􏼓ds ≤ e 􏽚 f(s)ds.
s 0
0
0
(2)
In [3], Kaijser et al. specifed that both (1) and (2) are
special cases of Hardy-Knopp’s inequality.
(3)
where Φ ∈ C(I, R), I⊆R is a convex function and
f: R+ ⟶ R+ is a locally integrable positive function.
In [4], Kaijser et al. applied Fubini’s theorem and Jensen’s inequality to establish an invitingly popularization of
(1). Particularly, it was evidenced that if ξ: (0, β) ⟶ R ≥ 0
and l: (0, β) × (0, β) ⟶ R ≥ 0, 0 < β ≤ ∞ such that
s
L(s) � 􏽚 l(s, t)dt > 0,
0
s ∈ (0, β),
(4)
and Φ ∈ C(I, R), I⊆R is a convex function and υ is defned
by
β
υ(t) � t 􏽚 ξ(s)
t
l(s, t) ds
< ∞,
L(s) s
Ten, the inequality,
t ∈ (0, β).
(5)
2
Journal of Mathematics
∞
􏽚 ξ(s)Φ Al f(s)􏼁
0
∞
ds
ds
≤ 􏽚 υ(s)Φ(f(s)) ,
s
s
0
holds for any non-negative integrable function
f: (0, β) ⟶ R such that f(s)⊆I where Al f is defned by
s
1
Al f(s) �
􏽚 l(s, t)f(t)dt,
L(s) 0
s ∈ (0, β).
(7)
As a popularization of (6), Krulic et al. [5] have dem􏽥 Σ,
􏽥 μ􏽥) are measure spaces
onstrated that if (Ξ, Σ, μ) and (Ξ,
with positive σ-fnite measures, ξ: Ξ ⟶ R ≥ 0 and l: Ξ ×
􏽥 ⟶ R ≥ 0 are measurable functions such that l(s, ·) is a d􏽥μ
Ξ
􏽥 L: Ξ ⟶ R is defned as
-integrable function for s ∈Ξ,
(q/p)
Al f(s)􏼁dμ(s)􏼓
􏼒􏽚 ξ(s)Φ
holds for any non-negative d􏽥μ-integrable function
􏽥 ⟶ R such that f(Ξ)⊆I
􏽥
f: Ξ
where Al f: Ξ ⟶ R is defned by
s ∈ Ξ.
(11)
In [6], the authors showed that if r ≥ 1, ξ: Ξ ⟶ R ≥ 0,
􏽥 ⟶ R ≥ 0 are measurable functions such that l(s, ·)
l: Ξ × Ξ
l(s, t) r−
Φ
􏽚 ξ(s)Φr Al f(s)􏼁dμ(s) + r􏽚 􏽚 ξ(s)
L(s)
Ξ
Ξ 􏽥
Ξ
1
s ∈ Ξ,
(q/p)
(q/p)
⎝􏽚 ξ(s)􏼠l(s, t)􏼡
υ(t) � ⎛
L(s)
Ξ
⎠
dμ(s)⎞
􏽥
t ∈Ξ,
< ∞,
(9)
where 0 < p ≤ q < ∞. If Φ is a convex function on I⊆R, then
the inequality,
(1/p)
≤ 􏼒􏽚 υ(s)Φ(f(s))d􏽥μ(s)􏼓
􏽥Ξ
(10)
,
􏽥 L: Ξ ⟶ R is defned by
is a d􏽥μ-integrable function for s ∈Ξ,
(8) and the function υ is defned as
r
l(s, t)
􏼡 dμ(s)􏼡
L(s)
υ(t) � 􏼠􏽚 ξ(s)􏼠
Ξ
(1/r)
< ∞,
􏽥
t ∈Ξ.
(12)
Moreover, if Φ is a non-negative superquadratic function, then
Φ is a convex function and ξ: Ξ ⟶ R+ ≥ 0 such that
υ(t) � 􏽚 ξ(s)
Ξ
l(s, t)
Δμ(s) < ∞,
l(s)
􏽥
t ∈ Ξ,
(15)
then
(14)
1
􏽚 ξ(t)Φ􏼠
􏽚 l(s, t)f(t)Δ􏽥μ(t)􏼡Δμ(s) ≤ 􏽚 υ(t)Φ(f(t))Δ􏽥μ(t),
􏽥Ξ
L(s) 􏽥Ξ
Ξ
􏽥 ⟶ R such
holds for all non-negative Δ􏽥μ-integrable f: Ξ
􏽥 ⊂ I.
that f(Ξ)
In [8], the authors improved the inequality (16) by
replacing the function f(t) by an m-tuple of functions.
f(t) � f1 (t), f2 (t), . . . , fm (t)􏼁,
(8)
r
􏼌􏼌
􏼌􏼌
Al f(s)􏼁 × Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑dμ(s)d􏽥μ(t) ≤ 􏼒􏽚 υ(t)Φ(f(t))d􏽥μ(t)􏼓 ,
􏽥Ξ
(13)
holds for any non-negative d􏽥μ-integrable function
􏽥 ⟶ R where Al f: Ξ ⟶ R is defned by (11).
f: Ξ
In [7], the researchers demonstrated some Hardy-type
inequalities with a general kernel. Tey have determined that
􏽥 Σ,
􏽥 μ􏽥) are two time scale measure spaces
if (Ξ, Σ, μ) and (Ξ,
􏽥 ⟶ R is such that
with positive σ-fnite measures, l: Ξ × Ξ
L(s) � 􏽚 l(s, t)Δ􏽥μ(t) < ∞,
􏽥Ξ
s ∈ Ξ,
and υ is defned by
(1/q)
Ξ
1
Al f(s) �
􏽚 l(s, t)f(t)d􏽥μ(t),
L(s) 􏽥Ξ
L(s) � 􏽚 l(s, t)d􏽥μ(t) < ∞,
􏽥Ξ
(6)
(17)
(16)
􏽥 as
such as f1 (t), f2 (t), . . . , fm (t) are Δ􏽥μ-integrable on Ξ
􏽥 ⟶ R are non-negative
follows: let ξ: Ξ ⟶ R, l: Ξ × Ξ
􏽥
such that l(s, ·) is a Δ􏽥μ-integrable function for s ∈Ξ,
L: Ξ ⟶ R and the function υ are defned by (14), (15),
respectively. Ten, for a convex function Φ over a convex set
I ⊂ Rm , the integral inequality,
Journal of Mathematics
3
􏽚 ξ(t)Φ􏼠
Ξ
1
􏽚 l(s, t)f(t)Δ􏽥μ(t)􏼡Δμ(s) ≤ 􏽚 υ(t)Φ(f(t))Δ􏽥μ(t),
􏽥Ξ
L(s) 􏽥Ξ
􏽥 ⟶ Rm such that
holds for all Δ􏽥μ-integrable functions f: Ξ
m
􏽥
f(Ξ) ⊂ U ⊂ R .
In [9], the authors derived some inequalities of Hardytype by utilizing the concept of superquadratic functions.
Particularly, they proved that if ξ: Ξ ⟶ R ≥ 0 and l: Ξ ×
(18)
􏽥 ⟶ R ≥ 0 are measurable functions such that l(s, ·) is a Δ􏽥μ
Ξ
􏽥 L: Ξ ⟶ R and the function υ
-integrable function for s ∈Ξ,
are defned by (14) and (15), respectively. Let Φ be a nonnegative superquadratic function. Ten, the inequality,
􏼌􏼌
l(s, t) 􏼌􏼌􏼌
Φ􏼐􏼌f(t) − Al f(s)􏼌􏼌􏼑Δμ(s)Δ􏽥μ(t) ≤ 􏽚 υ(t)Φ(f(t))Δ􏽥μ(t),
􏽚 ξ(s)Φ Al f(s)􏼁Δμ(s) + 􏽚 􏽚 ξ(s)
􏽥
􏽥Ξ
L(s)
Ξ
Ξ Ξ
holds for all non-negative Δ􏽥μ-integrable
􏽥 ⟶ R, where Al f: Ξ ⟶ R is defned by
f: Ξ
Al f(s) �
1
􏽚 l(s, t)f(t)Δ􏽥μ(t),
L(s) 􏽥Ξ
s ∈ Ξ.
function
(20)
(19)
Moreover, in [10, 11], the authors generalized (19) and
􏽥 ⟶ R ≥ 0 are
proved that if ξ: Ξ ⟶ R ≥ 0 and l: Ξ × Ξ
measurable functions such that l(s, ·) is a Δ􏽥μ-integrable
􏽥 L: Ξ ⟶ R and the function υ are defned
function for s ∈Ξ,
by (14) and (15), respectively. Let Φ be a non-negative
superquadratic function. Ten, the inequality
􏼌􏼌
l(s, t) 􏼌􏼌􏼌
Φ􏼐􏼌f(t) − Al f􏼁(s)􏼌􏼌􏼑Δμ(s)Δ􏽥μ(t) ≤ 􏽚 v(t)Φ(f(t))Δ􏽥μ(t),
􏽚 ξ(s)Φ Al f(s)􏼁Δμ(s) + 􏽚 􏽚 ξ(s)
􏽥Ξ
L(s)
Ξ
Ξ 􏽥
Ξ
􏽥 ⟶ Rm such that
holds for all Δ􏽥μ-integrable functions f: Ξ
m
􏽥
f(Ξ) ⊂ U ⊂ R , where Al f: Ξ ⟶ R is defned by
Al f(s) �
1
􏽚 l(s, t)f(t)Δ􏽥μ(t),
L(s) 􏽥Ξ
s ∈ Ξ.
(22)
In [12], the authors deduced several generalizations of
(19) on time scales. Tey proved that if r ≥ 1, ξ: Ξ ⟶ R ≥ 0
􏽥 ⟶ R ≥ 0 are measurable functions such that
and l: Ξ × Ξ
l(s, t) r−
Φ
􏽚 ξ(s)Φr Al f(s)􏼁Δμ(s) + r􏽚 􏽚 ξ(s)
L(s)
Ξ
Ξ 􏽥
Ξ
1
􏽥 L: Ξ ⟶ R is
l(s, ·) is a Δ􏽥μ-integrable function for s ∈Ξ,
defned by (14) and the function υ be defned by
r
l(s, t)
υ(t) � 􏼠􏽚 ξ(s)􏼠
􏼡 Δμ(s)􏼡
L(s)
Ξ
(1/r)
< ∞,
􏽥
t ∈Ξ.
(23)
Let Φ be a non-negative superquadratic function. Ten,
the inequality,
r
􏼌􏼌
􏼌􏼌
Al f(s)􏼁Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δμ(s)Δ􏽥μ(t) ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ(t)􏼓 , (24)
􏽥Ξ
holds for all non-negative Δ􏽥μ-integrable function
􏽥 ⟶ R where Al f: Ξ ⟶ R is defned by (20).
f: Ξ
Another development of Hardy-type inequality (24) has
been made by Saker et al. [13] as follows: let ξ: Ξ ⟶ R and
􏽥 ⟶ R be non-negative functions such that l(s, ·) is
l: Ξ × Ξ
􏽥 L: Ξ ⟶ R is defned by
a Δ􏽥μ-integrable function for s ∈Ξ,
0 < L(s) � 􏽚 l(s, t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 < ∞,
􏽥Ξ
and υ is defned by
(21)
s � s1 , s2 , . . . , sn 􏼁 ∈ Ξ,
(25)
4
Journal of Mathematics
(1/r)
r
l(s, t)
􏼡 Δμ s1 􏼁 . . . Δμ sn 􏼁􏼡
L(s)
υ(t) � 􏼠􏽚 ξ(s)􏼠
Ξ
􏽥
t � t1 , t2 , . . . , tm 􏼁 ∈Ξ,
< ∞,
(26)
where r ≥ 1. If Φ is a non-negative superquadratic function,
then the inequality,
l(s, t) r−
Φ
􏽚 ξ(s)Φr Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁 + r􏽚 􏽚 ξ(s)
L(s)
Ξ
Ξ 􏽥
Ξ
1
􏼌􏼌
􏼌􏼌
Al f(s)􏼁Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑
(27)
r
× Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 ,
􏽥Ξ
holds for all non-negative Δ􏽥μ-integrable
􏽥 ⟶ R where Al f: Ξ ⟶ R is defned by
f: Ξ
Al f(s) �
1
􏽚 l(s, t)f(t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁,
L(s) 􏽥Ξ
function
s ∈ Ξ.
(28)
In order to develop dynamic time scale inequalities, we
moved the reader to the articles [14–22].
Motivated by the previous results, our major aim in this
paper is to deduce several generalizations of general Hardytype inequalities for multivariate superquadratic functions
that involve more general kernels on arbitrary time scales.
Te paper is governed as follows: We remember some
basic notions, defnitions, and results of multivariate
superquadratic functions on time scales in Section 2. In
Section 3, we prove some new refned dynamic inequalities
of Hardy’s type with non-negative kernel by utilizing the
peculiarities of superquadratic (or subquadratic) functions.
In Section 4, we discuss several particular cases of Hardytype inequality by choosing such special kernels. Eventually,
in Section 5, we give more implementations of our obtained
results on particular time scales.
2. Preliminaries
In this section, we will introduce some fundamental concepts and efects to integrals of time scales and for multivariate superquadratic functions which will be useful to
deduce our major results.
Before introducing the main results for multidimensional inequalities, it is necessary to present some further
defnitions. Firstly, suppose m, n ∈ Z+ , Rm be the Euclidean
space. Let
s � s1 , s2 , . . . , sm 􏼁 ∈ Rm , t � t1 , t2 , . . . , tm 􏼁 ∈ Rm , f(t) � f1 (t), f2 (t), . . . , fm (t)􏼁,
be the function that is defned on t ∈ Rm . We utilize the
following notations:
s.t � s1 t1 , s2 t2 , . . . , sm tm 􏼁,
s
s s
s
� 􏼠 1 , 2 , . . . , m 􏼡,
t1 t2
tm
t
(29)
analogously, 0 � (0, 0, . . . , 0) ∈ Rm is the null vector and 1 �
(1, 1, . . . , 1) ∈ Rm . Correspondingly, for s, t ∈ Rm , s < t, we
defne (a, b) � {t ∈ Rm : a < t < b} and the n-cells [a, b),
(a, b] and [a, b] are defned similarly. Furthermore, the
subsets Km and K+m in Rm are defned by
Km � [0, ∞)m � 􏼈t ∈ Rm : 0 ≤ t􏼉, K+m � [0, ∞)m � 􏼈t ∈ Rm : 0 < t􏼉.
(31)
t
t
st � s11 s22 . . . stmm ,
(30)
􏼌􏼌 􏼌􏼌 􏼌􏼌 􏼌􏼌
􏼌􏼌 􏼌􏼌
|t| � 􏼐􏼌􏼌t1 􏼌􏼌, 􏼌􏼌t2 􏼌􏼌, . . . , 􏼌􏼌tm 􏼌􏼌􏼑,
m
⟨s, t⟩ � 􏽘 si ti ,
i�1
m
also, for s, t ∈ R , we write s ≤ t(s < t) if component wise
si ≤ ti (si < ti ), ∀1 ≤ i ≤ m, the relations ≥ , > rbin are defned
In particular,
m
m
2
⎝􏽙 t ⎞
⎠ and t−
t1 � 􏽙 ti , t2 � ⎛
i
i�1
i�1
m
1
− 1
⎝􏽙 t ⎞
⎠ ,
�⎛
i
(32)
i�1
where m � (m1 , m2 , . . . , mm ). Correspondingly, [a, b)
means
the
set
[a1 , b1 ) × [a2 , b2 ) × · · · × [am , bm ),
Δt � (Δt1 . . . Δtm ) and tp � (t1 . . . tm )p .
Journal of Mathematics
5
Now, we arraign the defnition and some essential
properties of superquadratic functions that are premised in
[23].
Example 2 (see [23]). Examples 4, 5, and 6 By utilizing the
same argument as in Example 4, the functions
Φ1 , Φ2 , Φ3 : Km ⟶ R that defne
m
Defnition 1 (see [23]). A function Φ: Km ⟶ R is called
superquadratic if for all s ∈ Km , there exists a function
c(s) ∈ Rm such that
Φ(t) − Φ(s) − Φ(|t − s|) ≥ ⟨c(s), t − s⟩,
∀t ∈ Km .
Φ1 (t) � 􏽘 ti cosh ti − sinh ti 􏼁,
i�1
m
(33)
i�1
Lemma 1 (see [25]). Suppose Φ: [0, ∞) ⟶ R is continuously diferentiable and Φ(0) ≤ 0. If Φ is superadditive or
′
Φ′ (x)/x is nondecreasing, then Φ is superquadratic.
In the next, we recall a couple of benefcial examples of a
superquadratic function.
⎧
⎪
2
⎪
⎨ 􏽘 ti ln ti ,
Φ3 (t) � ⎪ i�1,i ≠ j
⎪
⎩
0,
m
(34)
i�1
which is superquadratic on Km for each p ≥ 2 (as shown in
[23], Example 2) and the function Φ: Km ⟶ R is defned
by
m
(1/p)
p
⎝􏽘 t ⎞
⎠
Φ(t) � − ⎛
i
,
(35)
i�1
if ti > 0, tj � 0,
if t � 0,
are superquadratic.
Te following lemma shows that non-negative superquadratic functions are indeed convex functions:
Lemma 2 (see [23]). Let Φ: Km ⟶ R be a superquadratic
with c(t) � (c1 (t), c2 (t), . . . , cm (t)) that is defned as in
Defnition 1. Ten,
(i) Φ(0) ≤ 0 and ci (0) ≤ 0 ∀1 ≤ i ≤ m;
(ii) If Φ(0) � 0 and ∇Φ(0) � (z1 Φ(0), z2 Φ(0), . . . ,
zm Φ(0)) � 0, then ci (t) � zi Φ(t) whenever zi Φ(t)
exists for some index 1 ≤ i ≤ m at t ∈ Km ; where ∇Φ
means that the gradient of the function Φ and zi Φ(t)
denotes the partial derivative of Φ over the i-th
variable;
(iii) If Φ ≥ 0, then Φ is convex and Φ(0) � ∇Φ(0) � 0.
Example 1 (see [23]). Te power function Φ: Km ⟶ R
that is defned as Φ(t) � tp is called superquadratic if p ≥ 2
and subquadratic if 1 < p ≤ 2 (it is also readily seen that if
0 < p ≤ 1 then tp is a subquadratic function). Since the sum of
superquadratic functions is also superquadratic, the function
Φ: Km ⟶ R is defned as follows:
Φ(t) � 􏽘 tpi ,
(36)
i�1
m
If − Φ is superquadratic, then Φ is subquadratic and the
reverse inequality of (22) is held.
Defnition 2 (see [24]). A function Φ: [0, ∞) ⟶ R is
superadditive provided Φ(x + y) ≥ Φ(x) + Φ(y) for all
x, y ≥ 0. If the reverse inequality holds, then Φ is said to be
subadditive.
m
⎝1 + 􏽘 t ⎞
⎠ − 􏽘t ,
Φ2 (t) � ln⎛
i
i
The following defnitions and theorems are referred
from [26, 27]. Let T i , 1 ≤ i ≤ n be time scales, and
Λn � T 1 × T 2 × · · · × T n � 􏼈t � t1 , t2 , . . . , tn 􏼁: ti ∈ T i ,
1 ≤ i ≤ n􏼉,
(37)
which is called an n-dimensional time scale. Let E be Δ
-measurable subset of Λn and f: E ⟶ R be a Δ-measurable
function. Ten, the corresponding Δ-integral is called
Lebesgue Δ-integral and is denoted by
which is superquadratic on Km for each p ≥ 1 (see [23],
Example 3).
􏽚 f t1 , t2 , . . . , tn 􏼁Δ1 t1 . . . Δn tn , 􏽚 f(t)Δ1 t1 . . . Δn tn , 􏽚 fdμΔ or 􏽚 f(t)dμΔ (t),
E
E
where μΔ is a σ-additive Lebesgue Δ-measure on Λn . Also, if
f(t) � f1 (t), f2 (t), . . . , fm (t)􏼁,
(39)
which is an m-tuple of functions in n-variables such that
f1 , f2 , . . . , fm are Lebesgue Δ-integrable on E, then 􏽒E fdμΔ
denotes the m-tuple.
􏼒􏽚 fdμΔ 1 , . . . , 􏽚 fm dμΔ 􏼓,
E
E
(40)
E
(38)
E
i.e., Δ-integral acts on each component of f.
Particularly, if T is an arbitrary time scale and [s, t) ⊂ T
includes only isolated points, then
t
􏽚 f(θ)Δθ � 􏽘 (σ(θ) − θ)f(θ) � 􏽘 f(θ)μ(θ),
s
θ∈[s,t)
(41)
θ∈[s,t)
where μ is referred to as the graininess function on time
scale.
6
Journal of Mathematics
Lemma 3 (Minkowski’s inequality [13]). Let (Ξ, Σ, μ) and
􏽥 Σ,
􏽥 μ􏽥) be two fnite-dimensional time scale measures
(Ξ,
(1/λ)
λ
􏼠􏽚 􏼒􏽚 f(s, t)υ(t)d􏽥μ(t)􏼓 ξ(s)dμ(s)􏼡
􏽥Ξ
Ξ
provided that all integrals in (23) exist. If 0 < λ < 1,
λ
(43)
􏽚 􏼒􏽚 fυd􏽥μ􏼓 ξdμ > 0 and 􏽚 fυd􏽥μ > 0,
􏽥Ξ
􏽥Ξ
Ξ
then (42) is reversed. For λ < 0, if in addition to (43) and
􏽚 fλ ξdμ > 0,
(44)
Ξ
then (42) is again reversed.
⎝
Φ⎛
􏽒E wfdμΔ
􏽒E wdμΔ
⎠≤
⎞
spaces. Suppose that ξ ≥ 0, v ≥ 0 and f ≥ 0 that are defned on
􏽥 and Ξ × Ξ,
􏽥 respectively. If λ ≥ 1, then
Ξ, Ξ
≤ 􏽚 􏼒􏽚 fλ (s, t)ξ(s)dμ(s)􏼓υ(t)d􏽥μ(t),
􏽥Ξ Ξ
In [25], (Corollary 5.1), Bibi get the following generalization of Jensen’s inequality and the converse of it for
superquadratic functions.
Theorem 1 (Jensen’s inequality). Let Φ ∈ C(Km , R) be a
superquadratic function and w be Δ-integrable function for
w ≥ 0 and 􏽒E wdμΔ . Ten, for every Δ-integrable functions f
such that f(E) ⊂ Km and wf, wΦ(f) are Δ-integrable, we
have
􏼌􏼌
􏼌􏼌
􏽒E wΦ(f)dμΔ − 􏽒E wΦ􏼒􏼌􏼌􏼌f − 􏼐􏽒E wfdμΔ /􏽒E wdμΔ 􏼑 · 1􏼌􏼌􏼌􏼓dμΔ
􏽒E wdμΔ
.
(45)
􏽥 ⟶ R be a non-negative so that l(s, ·) is
(B2 )l: Ξ × Ξ
􏽥 and L: Ξ ⟶ R be
Δ􏽥μ-integrable function for s ∈Ξ
defned by
If Φ is subquadratic, then (45) is reversed.
3. Main Results
In this section, we prove multidimensional Hardy-type
inequalities with general kernels on time scales. Before
proceeding with results, we introduce the following
notations:
􏽥 Σ,
􏽥 μ􏽥) are two time scale delta
(B1 )(Ξ, Σ, μ) and (Ξ,
measure spaces with positive σ-fnite measures.
0 < L(s) � 􏽚 l(s, t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 < ∞,
􏽥Ξ
(B3 )ξ: Ξ ⟶ R is Δμ-integrable and the function υ be
defned by
r
l(s, t)
􏼡 Δμ s1 􏼁 . . . Δμ sn 􏼁􏼡
L(s)
where r ≥ 1.
s ∈ Ξ.
(46)
(1/r)
υ(t) � 􏼠􏽚 ξ(s)􏼠
Ξ
(42)
< ∞,
􏽥
t ∈Ξ,
(47)
Theorem 2. Weassume (B1 ) − (B3 ) are satisfed. If
Φ ∈ C(Km , R) is non-negative superquadratic function, then
l(s, t) r−
Φ
􏽚 ξ(s)Φr Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁 + r􏽚 􏽚 ξ(s)
􏽥Ξ Ξ
L(s)
Ξ
1
􏼌􏼌
􏼌􏼌
Al f(s)􏼁Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑
(48)
r
× Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 ,
􏽥Ξ
􏽥 ⟶ Rm such that
holds for all Δ􏽥μ-integrable functions f: Ξ
􏽥 ⊂ Km ⊂ Rm , where Al f: Ξ ⟶ R is defned by
f(Ξ)
Al f(s) �
1
􏽚 l(s, t)f(t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁,
L(s) 􏽥Ξ
s ∈ Ξ. (49)
Journal of Mathematics
7
By applying (45) on (50), we fnd
If 0 < r < 1 and Φ is subquadratic, then (48) is reversed.
Proof. We begin with the following identity:
1
Φ Al f(s)􏼁 � Φ􏼠
􏽚 l(s, t)f(t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡.
L(s) 􏽥Ξ
(50)
Φ Al f(s)􏼁 +
􏼌􏼌
􏼌􏼌
1
1
􏽚 l(s, t)Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 ≤
􏽚 l(s, t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁.
L(s) 􏽥Ξ
L(s) 􏽥Ξ
(51)
Since Φ ≥ 0 and r ≥ 1, we have
􏼠Φ Al f(s)􏼁 +
r
r
􏼌􏼌
􏼌􏼌
1
1
􏽚 l(s, t)Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡 ≤ 􏼠
􏽚 l(s, t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡 .
L(s) 􏽥Ξ
L(s) 􏽥Ξ
Furthermore, by employing the Bernoulli inequality
[28],
(1 + x)c ≤ 1 + cx,
c
cx + 1 − c ≤ x ,
for 0 < c ≤ 1 and x > − 1,
for c ≥ 1 and x > 0.
Φr Al f(s)􏼁 + r
Φr−
(52)
It follows that the L. H. S of (52) is not less than
(53)
1
􏼌􏼌
􏼌􏼌
Al f(s)􏼁
􏽚 l(s, t)Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁,
􏽥Ξ
L(s)
(54)
that is,
Φr Al f(s)􏼁 + r
Φr−
r
􏼌􏼌
􏼌􏼌
Al f(s)􏼁
1
􏽚 l(s, t)Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 ≤ 􏼠
􏽚 l(s, t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡 .
􏽥Ξ
L(s)
L(s) 􏽥Ξ
1
(55)
By multiplying (55) with ξ(s) and integrating it over Ξ
with respect to Δμ(s1 ) . . . Δμ(sn ), we get
􏽚 ξ(s)Φr Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁
Ξ
+ r􏽚
􏼌􏼌
􏼌􏼌
ξ(s)Φr− 1 Al f(s)􏼁
􏼒􏽚 l(s, t)Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑Δ􏽥μ t1 􏼁 · · · μ􏽥 tn 􏼁􏼓 × Δμ s1 􏼁 . . . Δμ sn 􏼁
􏽥
L(s)
Ξ
Ξ
r
1
≤ 􏽚 ξ(s)􏼠
􏽚 l(s, t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡 Δμ s1 􏼁 . . . Δμ sn 􏼁.
L(s) 􏽥Ξ
Ξ
Applying (42) on the R. H. S of (56), we get
(56)
8
Journal of Mathematics
r
􏽚 ξ(s)􏼠
Ξ
1
􏽚 l(s, t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼡 Δμ s1 􏼁 . . . Δμ sn 􏼁
L(s) 􏽥Ξ
r
⎝􏽚 Φ(f(t))􏼠􏽚 ξ(s)􏼠l(s, t)􏼡 Δμ s 􏼁 . . . Δμ s 􏼁􏼡
≤⎛
1
n
􏽥Ξ
L(s)
Ξ
(57)
r
(1/r)
⎠.
Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁⎞
Finally, substituting (57) into (56) and using the defnition (47) of the weight function υ, we have
􏽚 ξ(s)Φr Al f(s)􏼁Δμ1 s1 􏼁 . . . Δμ1 sn 􏼁
Ξ
l(s, t) r−
Φ
+ r􏽚 􏽚 ξ(s)
􏽥Ξ Ξ
l(s)
1
􏼌􏼌
􏼌􏼌
Al f(s)􏼁Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑 × Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁
(58)
r
≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 .
􏽥Ξ
Tis proves (48). Te proof of the case in which 0 < r < 1
and Φ is subquadratic is similar; the only diference is that
the inequality sign in (48) is reversed. Te proof is
complete.
□
Remark 1. For m � 1, inequality (48) in Teorem 2 reduces
to (27).
Remark 2. For n � 1 and r � 1, inequality (48) in Teorem 2
coincides with (21).
Corollary 1. Given that ξ and Al f(s) are as in Teorem 2
and ω ≥ 0 is measurable function. Since Φ ≥ 0 is superquadratic function, then the second term on the L. H. S of (48)
is non-negative and the integral inequality
r
􏽚 ξ(s)Φr Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁 ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 ,
􏽥Ξ
Ξ
holds.
Remark 3. By taking r � 1 and n � 1, inequality (59) in
Corollary 1 reduces to (18).
(59)
Remark 4. For m � 1 and n � 1, inequality (59) in Corollary
1 coincides with inequality (2.2) which is [29], (Corollary
2.1.2).
Remark 5. If we rewrite (48) with r � λ/μ ≥ 1, 0 < μ ≤ λ < ∞
or − ∞ < μ ≤ λ < 0 and Φ ≥ 0, then we get
􏽚 ξ(s)Φ(λ/μ) Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁
Ξ
λ
l(s, t) (λ/μ)−
Φ
+ 􏽚 􏽚 ξ(s)
μ Ξ 􏽥Ξ
L(s)
1
􏼌􏼌
􏼌􏼌
Al f(s)􏼁Φ􏼐􏼌􏼌f(t) − Al f(s)􏼌􏼌􏼑
(60)
(λ/μ)
× Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁 ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓
􏽥Ξ
Remark 6. For m � 1 and n � 1, inequality (60) coincides
with inequality (3.13) which is [27], (Remark 3.5).
.
Remark 7. In Remark 5, since Φ ≥ 0, then the second
term on the L. H. S. of (60) is non-negative and (60)
Journal of Mathematics
9
reduces to the weighted Hardy-type inequality of the
form
􏽚 ξ(s)Φ(λ/μ) Al f(s)􏼁Δμ s1 􏼁 . . . Δμ sn 􏼁 ≤ 􏼒􏽚 υ(t)Φ(f(t))Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓
􏽥Ξ
Ξ
which is a refnement of general Hardy-type inequality
established in [29] [Remark 2.1.4] and [5].
As a specifc case of Teorem 2 when Φ(t) � tλ for λ ≥ 2,
we get the next result.
(λ/μ)
(61)
,
Corollary 2. Suppose that the assumptions of Teorem 2 are
satisfed and λ ≥ 2. Ten,
􏼌􏼌λ
l(s, t)
λr
λ(r− 1) 􏼌􏼌􏼌
Al f(s)􏼁
􏽚 ξ(s) Al f(s)􏼁 Δμ s1 􏼁 . . . Δμ sn 􏼁 + r􏽚 􏽚 ξ(s)
􏼌f(t) − Al f(s)􏼌􏼌 × Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁
L(s)
Ξ
Ξ 􏽥
Ξ
λ
(62)
r
≤ 􏼒􏽚 υ(t)f (t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 .
􏽥Ξ
If 0 < r < 1 and 1 < λ ≤ 2, then (62) is reversed.
Remark 8. Clearly, for m � 1, inequality (62) in Corollary 2
coincides with inequality (46) which is [13], (Corollary 2.1).
In fact, the function Φ(s) � es is not superquadratic but
by working with the superquadratic function Φ(s) � es −
s − 1 (see Lemma 3) and replacing f(s) by ln f(s) in Teorem
2, we obtain the next multidimensional version of the
Pólya–Knopp type inequality.
Corollary 3. Suppose that the assumptions in Teorem 2 are
satisfed and assume that
Al f(s) �
1
􏽚 l(s, t)ln f(t)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁,
L(s) 􏽥Ξ
s ∈ Ξ,
(63)
then
r
r
􏽚 ξ(s) exp Al f(s) − Al f(s) − 1􏼁 Δμ s1 􏼁 . . . Δμ sn 􏼁 + I ≤ 􏼒􏽚 υ(t)(f(t) − ln f(t) − 1)Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁􏼓 ,
􏽥Ξ
Ξ
(64)
l(s, t)
r− 1
I � r􏽚 􏽚 ξ(s)
exp Al f(s) − Al f(s) − 1􏼁
􏽥
L(s)
Ξ Ξ
􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌
× 􏼐exp􏼌􏼌ln f(t) − Al f(s)􏼌􏼌 − 􏼌􏼌lnf(t) − Al f(s)􏼌􏼌 − 1􏼑 × Δμ s1 􏼁 . . . Δμ sn 􏼁Δ􏽥μ t1 􏼁 . . . Δ􏽥μ tn 􏼁.
(65)
where
If 0 < r < 1, then (64) is reversed.
Remark 9. For m � 1, inequality (64) in Corollary 3 coincides with inequality (48) which is [13], (Corollary 2.2).
In the next results, we also suppose the following
hypothesis:
(B1′)
Let
􏽥 � [a1 , β1 )T × [a2 , β2 )T × · · · × [an , βn )T ⊂ Rn ,
Ξ�Ξ
0 ≤ ai < βi ≤ ∞, for every 1 ≤ i ≤ n, where T is an arbitrary
time scale and let Δμ1 (s) � Δs, Δμ2 (t) � Δt.
Theorem 3. Weassume (B1′) and (B2 ) are satisfed. Suppose
τ: Ξ ⟶ R+ such that
r
ω(t) � s1 . . . sn 􏼁􏼠􏽚
1
l(s, t)
τ(s)􏼠
􏼡 Δs1 . . . Δsn 􏼡
L(s)
Ξ σ s 1 􏼁 . . . σ sn 􏼁
(1/r)
,
t ∈ Ξ,
(66)
10
Journal of Mathematics
where r ≥ 1. If Φ ∈ C(Km , R) is non-negative superquadratic
function, then
β1
βn
a1
an
􏽚 . . . 􏽚 τ(s)Φr A∗l f(s)􏼁
β1
βn
a1
an
β1
βn
t1
tn
Δs1 . . . Δsn
σ s1 􏼁 . . . σ s n 􏼁
+ r 􏽚 . . . 􏽚 􏽚 . . . 􏽚 τ(s)
l(s, t) r−
Φ
L(s)
1
􏼌􏼌
􏼌􏼌
A∗l f(s)􏼁Φ􏼐􏼌􏼌f(t) − A∗l f(s)􏼌􏼌􏼑
(67)
r
×
β1
βn
Δs1 . . . Δsn
Δt1 . . . Δtn ≤ 􏼠􏽚 . . . 􏽚 ω(t)Φ(f(t))Δt1 . . . Δtn 􏼡 ,
σ s1 􏼁 . . . σ s n 􏼁
a1
an
holds for all Δ􏽥μ-integrable functions f: Ξ ⟶ Rm such that
f(Ξ) ⊂ Km ⊂ Rm , where A∗l f: Ξ ⟶ R is defned by
A∗l f(s) �
β1
βn
1
􏽚 . . . 􏽚 l(s, t)f(t)Δt1 . . . Δtn ,
L(s) a1
an
Remark 10. For r � 1, Teorem 3 coincides with [10],
(Corollary 2.11).
s ∈ Ξ.
Remark 11. In Teorem 3, if we replace ω(t) by
ω(t)/(s1 . . . sn ) and put m � 1, then we get the result given in
[13], (Corollary 2.3).
(68)
If Φ is subquadratic function and 0 < r < 1, then (48) is
reversed.
4. Inequalities with Special Kernels
In this section, we get some consequential inequalities of
Hardy-type by selecting special kernels.
Proof. We get the result from Teorem 2 by taking
ξ(s) �
τ(s)
.
σ s 1 􏼁 . . . σ sn 􏼁
(69)
□
β1
βn
t1
tn
ω2 (t) � 􏼠􏽚 . . . 􏽚 ξ(s)􏼠
Theorem . Suppose (B1′) and
1
r
􏽑ni�1
σ si 􏼁 − ai 􏼁􏼁 Δs1 . . . Δsn )(1/r)
< ∞,
(70)
such that ξ: Ξ ⟶ R+ is Δμ-integrable function and r ≥ 1. If
Φ ∈ C(Km , R) is non-negative superquadratic function, then
β1
βn
a1
an
􏽚 . . . 􏽚 ξ(s)Φr Al∗ ∗ f(s)􏼁Δs1 . . . Δsn
β1
βn
a1
an
β1
βn
t1
tn
+ r􏽚 ...􏽚 􏽚 ...􏽚
􏽑ni�1
ξ(s)
Φr−
σ si 􏼁 − a i 􏼁
β1
βn
a1
an
1
􏼌
􏼌
Al∗ ∗ f(s)􏼁Φ􏼐􏼌􏼌f(t)
r
≤ 􏼠􏽚 . . . 􏽚 ω2 (t)Φ(f(t))Δt1 . . . Δtn 􏼡 ,
holds for all Δ􏽥μ-integrable functions f: Ξ ⟶ Rm such that
f(Ξ) ⊂ Km ⊂ Rm , where Al∗ ∗ f: Ξ ⟶ R is defned by
−
􏼌
􏼌
Al∗ ∗ f(s)􏼌􏼌􏼑
(71)
× Δs1 . . . Δsn Δt1 . . . Δtn
Journal of Mathematics
11
Al∗ ∗ f(s) �
σ (s1 )
σ (sn )
1
...􏽚
f(t)Δt1 . . . Δtn ,
􏽚
σ si 􏼁 − a i 􏼁 a 1
an
􏽑ni�1
s ∈ Ξ.
(72)
If Φ is subquadratic function and 0 < r < 1, then (71) is
reversed.
Remark 12. For r � 1, Teorem 4 coincides with [10],
(Corollary 3.1).
Proof. We get the result from Teorem 2 by taking
We have the following in this case:
Remark 13. For m � 1, Teorem 4 coincides with [13],
(Corollary 2.7).
If we let ai � 0, (i � 1, . . . , n) and ξ(s) � 1/(s1 . . . sn ) in
Teorem 4, we have the next result.
L(s) � 􏽚
σ (s 1 )
a1
σ (s n )
...􏽚
an
n
Δt1 . . . Δtn � 􏽙 σ si 􏼁 − ai 􏼁, (73)
i�1
Al � Al∗ ∗ , ω � ω2 .
□
β1
βn
t1
tn
ω3 (t) � 􏼠􏽚 . . . 􏽚
Corollary . Assuming (B1′), we defne
1
r Δs1 . . . Δsn 􏼡
s1 . . . sn 􏼁 􏽑ni�1 σ si 􏼁􏼁
(1/r)
,
t ∈ Ξ,
(74)
such that ξ: Ξ ⟶ R+ is Δμ-integrable function and r ≥ 1. If
Φ ∈ C(Km , R) is non-negative superquadratic function, then
β1
βn
0
0
􏽚 . . . 􏽚 Φr A′lf(s)􏼁
Δs1 . . . Δsn
􏽑ni�1 si
β1
βn
β1
βn
0
0
t1
tn
+ r 􏽚 . . . 􏽚 􏽚 . . . 􏽚 Φr−
1
􏼌􏼌
􏼌􏼌
A′lf(s)􏼁Φ􏼐􏼌􏼌f(t) − A′lf(s)􏼌􏼌􏼑
(75)
r
×
β1
βn
Δs1 . . . Δsn
Δt . . . Δtn ≤ 􏼠􏽚 . . . 􏽚 ω3 (t)Φ(f(t))Δt1 . . . Δtn 􏼡 ,
􏽑ni�1 si σ si 􏼁 1
0
0
holds for all Δ􏽥μ-integrable functions f: Ξ ⟶ Rm such that
f(Ξ) ⊂ Km ⊂ Rm , where A′lf: Ξ ⟶ R is defned by
A′lf(s) �
σ (s1 )
σ (sn )
1
􏽚
...􏽚
f(t)Δt1 . . . Δtn ,
σ si 􏼁 0
0
If Φ is subquadratic function and 0 < r < 1, then (76) is
reversed.
s ∈ Ξ.
􏽑ni�1
Remark 14. If we take βi � ∞(i � 1, . . . , n), then inequalities (75) and (76) reduces to
(76)
∞
(1/r)
∞
ω3 (t) � 􏼠􏽚 . . . 􏽚
t1
∞
∞
0
0
1
r Δs1 . . . Δsn 􏼡
s1 . . . sn 􏼁 􏽑ni�1 σ si 􏼁􏼁􏼁
tn
􏽚 . . . 􏽚 Φr A′lf(s)􏼁
∞
0
t ∈ Ξ,
Δs1 . . . Δsn
􏽑ni�1 si
∞
∞
∞
0
t1
tn
+ r 􏽚 . . . 􏽚 􏽚 . . . 􏽚 Φr−
∞
,
1
(77)
􏼌􏼌
􏼌􏼌
Δs . . . Δsn
A′lf(s)􏼁Φ􏼐􏼌􏼌f(t) − AA′lf(s)􏼌􏼌􏼑 × n1
Δt . . . Δtn
􏽑i�1 si σ si 􏼁 1
∞
r
≤ 􏼒􏽚 . . . 􏽚 ω3 (t)Φ(f(t))Δt1 . . . Δtn 􏼓 .
0
0
Remark 15. Clearly, for r � 1, inequalities (75) and (76),
respectively, reduces to
β1
βn
t1
tn
ω3 (t) � 􏽚 . . . 􏽚
n
1
1 1
Δs
.
.
.
Δs
�
􏽙
􏼠 − 􏼡,
n
1
n
ti βi
􏽑i�1 si σ si 􏼁
i�1
(78)
12
Journal of Mathematics
and
β1
βn
0
0
􏽚 ...􏽚
1
Φ A′lf(s)􏼁Δs1 . . . Δsn
􏽑ni�1 si
β1
βn
β1
βn
0
0
t1
tn
β1
βn
n
0
0 i�1
+ 􏽚 ...􏽚 􏽚 ...􏽚
􏼌􏼌
􏼌􏼌
1
1 1
Φ􏼐􏼌􏼌f(t) − A′lf(s)􏼌􏼌􏼑Δs1 . . . Δsn Δt1 . . . Δtn 􏼠 − 􏼡Φ(f(t))Δt1 . . . Δtn .
􏽑ni�1 si σ si 􏼁
ti βi
(79)
≤ 􏽚 ...􏽚 􏽙
Furthermost, if we take βi � ∞(i � 1, . . . , n), then (80)
becomes
∞
∞
􏽚 ...􏽚
0
∞
1
n
0 􏽑i�1 si
∞
∞
∞
Φ A′lf(s)􏼁Δs1 . . . Δsn + 􏽚 . . . 􏽚 􏽚 . . . 􏽚
0
0
t1
tn
∞
∞ n
􏼌􏼌
􏼌􏼌
1
Φ􏼐􏼌􏼌f(t) − A′lf(s)􏼌􏼌􏼑 × Δs1 . . . Δsn Δt1 . . . Δtn ≤ 􏽚 . . . 􏽚 􏽙 ti ,
n
􏽑i�1 si σ si 􏼁
0
0 i�1
Theorem 5. Assume (B1′) with βi � ∞(i � 1, . . . , n). If
Φ ∈ C(Km , R) is non-negative superquadratic function, then
which is [10], [Remark 3.2].
∞
∞
n
a1
an
i�1
∞
∞
s1
sn
⎝􏽙 s 􏽚 . . . 􏽚
􏽚 . . . 􏽚 Φr ⎛
i
∞
∞
σ (t 1 )
+ r􏽚 ...􏽚 􏽚
a1
(80)
an
σ (t n )
...􏽚
a1
an
f(t)
⎠ Δs1 .n. . Δsn
Δt . . . Δtn ⎞
􏽑ni�1 ti σ ti 􏼁 1
􏽑i�1 si
n
∞
∞
s1
sn
⎝􏽙 s 􏽚 . . . 􏽚
Φr− 1 ⎛
i
i�1
f(t)
⎠
Δt . . . Δtn ⎞
􏽑ni�1 ti σ ti 􏼁 1
􏼌􏼌
􏼌􏼌
n
∞
∞
􏼌􏼌
􏼌􏼌
f(t)
􏼌
⎝
⎞
× Φ⎛􏼌􏼌􏼌f(t) − 􏽙 si 􏽚 . . . 􏽚
Δt1 . . . Δtn 􏼌􏼌􏼌􏼌⎠
n
􏼌
􏼌
s1
sn 􏽑i�1 ti σ ti 􏼁
i�1
(81)
r
× Δs1 . . . Δsn
∞
∞
Δt1 . . . Δtn
≤ 􏼠􏽚 . . . 􏽚 ω(t)Φ(f(t))Δt1 . . . Δtn 􏼡 ,
n
􏽑i�1 ti σ ti 􏼁
a1
an
holds for all Δ􏽥μ-integrable functions f: Ξ ⟶ Rm such that
f(Ξ) ⊂ Km ⊂ Rm , where
∞
∞
a1
n
an 􏽑i�1 si
⎝􏽚 . . . 􏽚
ω(t) � ⎛
1
⎝􏼠
⎛
1/􏽑ni�1 ti σ ti 􏼁􏼁
􏼡
r
1/􏽑ni�1 si 􏼁􏼁 Δs1 . . . Δsn
n
If Φ is subquadratic function and 0 < r < 1, then (82) is
reversed.
Proof. We get the result from Teorem 2 by taking
l(s, t) � 􏼨 􏽙
i�1
(1/r)
,
t ∈ Ξ.
(82)
1
, if si ≥ t for all i ∈ {1, . . . , n}, 0, otherwise.
t i σ ti 􏼁
(83)
and ξ(s) � 1/(s1 . . . sn ).
□
Journal of Mathematics
13
Corollary 5. Clearly, for r � 1, inequality (82) reduces to
∞
∞
a1
an
n
∞
∞
s1
sn
⎝􏽙 s 􏽚 . . . 􏽚
􏽚 . . . 􏽚 Φ⎛
i
i�1
∞
σ (t 1 )
∞
+ 􏽚 ...􏽚 􏽚
a1
an
× Δs1 . . . Δsn
f(t)
⎠ Δs1 .n. . Δsn
Δt . . . Δtn ⎞
Πi�1 si
􏽑ni�1 ti σ ti 􏼁 1
σ (t n )
...􏽚
a1
an
􏼌􏼌
􏼌􏼌
n
∞
∞
􏼌􏼌
􏼌􏼌
f(t)
􏼌
⎝
⎠
⎛
Δt1 . . . Δtn 􏼌􏼌􏼌􏼌⎞
Φ 􏼌􏼌􏼌f(t) − 􏽙 si 􏽚 . . . 􏽚
n
􏼌
􏼌
s1
sn 􏽑i�1 ti σ ti 􏼁
i�1
(84)
∞
∞ n
Δt1 . . . Δtn
ai
Δt . . . Δtn
≤ 􏽚 . . . 􏽚 􏽙􏼠1 −
,
􏼡Φ(f(t)) 1 n
n
σ
t
􏽑i�1 ti σ ti 􏼁
􏽑i�1 ti
􏼁
a1
an i�1
i
Theorem 6. Assume T be an isolated time scale, ξ: Ξ ⟶ R+
􏽥 � [a, ∞)T , a ≥ 0. If λ ≥ 1,
is Δμ-integrable function and Ξ � Ξ
then
which is [10], (Corollary 3.3).
5. Inequalities with Particular Time Scales
In this section, we get some consequential inequalities by
selecting some specifc time scales.
λ
(r/λ)
λ
ξ(s)􏼔 A″l f1 􏼁 (s) + A″l f2 􏼁 (s)􏼕
􏽘
μ(s) + r
s∈[a,∞)T
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
(r− 1/λ)
ξ(s)
λ
λ
􏼔 A″l f1 􏼁 (s) + A″l f2 􏼁 (s)􏼕
σ(s) − a
􏼌􏼌λ (1/λ)
􏼌􏼌
􏼌􏼌λ 􏼌􏼌􏼌
′′
􏼌
μ(s)μ(t)
􏼠􏼌􏼌f1 (t) − A″l f1 (s)􏼌􏼌 + 􏼌􏼌􏼌f2 (t) − Al f2 (s)􏼌􏼌􏼌 􏼡
(85)
r
ω4 (t)􏼐fλ1 (t)
⎝ 􏽘
≥⎛
+
(1/λ)
⎠
fλ2 (t)􏼑
μ(t)⎞
,
t ∈ [a,∞)T
􏽥 ⟶ R+ , where
holds for all Δ􏽥μ-integrable functions f1 , f2 : Ξ
1
A″l fi 􏼁(s) �
σ(s) − a
fi (t)μ(t),
􏽘
i � 1, 2,
(86)
Proof. We get the result from Teorem 4 by taking
(1/λ)
m
⎝ 􏽘 sλ ⎞
⎠
Φ(s) � − ⎛
i
t∈[a,σ(s))T
(88)
,
i�1
and
□
with λ ≥ 1, n � 1 and m � 2.
⎝
ω4 (t) � ⎛
􏽘
s ∈ [t,∞)T
􏽘
′
λ
ξ(s)
⎠
μ(s)⎞
(σ(s) − a)r
′
λ
ξ(s)􏼢􏼒A′l f1 􏼓 (s) + 􏼒A′l f2 􏼓 (s)􏼣
(1/r)
.
(87)
Remark 16. If we take r � 1 in Teorem 6, then we get
(1/λ)
μ(s)
s∈[a,∞)T
+
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
􏼌􏼌λ 􏼌􏼌
􏼌􏼌λ (1/λ)
ξ(s) 􏼌􏼌􏼌
μ(s)μ(t) ≥
􏼒􏼌f1 (t) − A″l f1 (s)􏼌􏼌 + 􏼌􏼌f2 (t) − A″l f2 (s)􏼌􏼌 􏼓
σ(s) − a
􏽘
ω5 (t)􏼐fλ1 (t) + fλ2 (t)􏼑
(1/λ)
μ(t),
t∈[a,∞)T
(89)
where (A″l fi )(s), i � 1, 2 is defned by (87) and
14
Journal of Mathematics
ω5 (t) �
􏽘
s∈[t,∞)T
ξ(s)
μ(s),
σ(s) − a
(90)
Theorem 7. Assuming T be an isolated time scale,
􏽥 � [a, ∞)T ,
ξ: Ξ ⟶ R+ is Δμ-integrable function and Ξ � Ξ
a ≥ 0. Ten,
which is [10], (Corollary 4.1).
∧
∧
∧
∧
∧
∧
ξ(s)μ(s)􏼒􏼒Al f1 􏼓(s)cosh􏼒Al f1 􏼓(s) − sinh􏼒Al f1 􏼓(s)+􏼓 + 􏼒Al f2 􏼓(s)cosh􏼒Al f2 􏼓(s) − sinh􏼒Al f2 􏼓(s)r + r
􏽘
s∈[a,∞)T
r−
∧
∧
∧
∧
∧
∧
ξ(s)
􏼒􏼒Al f1 􏼓(s)cosh􏼒Al f1 􏼓(s) − sinh􏼒Al f1 􏼓(s) + 􏼒Al f2 􏼓(s)cosh􏼒Al f2 􏼓(s) − sinh􏼒Al f2 􏼓(s)􏼓
σ(s) − a
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
1
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
∧
∧
∧
􏼌􏼌
􏼌􏼌
􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌
􏼌
􏼌
􏼌
􏼌
× 􏼒􏼌􏼌f1 (t) − 􏼒Al f1 􏼓 (s)􏼌􏼌cosh􏼌􏼌f1 (t) − 􏼒Al f1 􏼓(s)􏼌􏼌 − sinh􏼌􏼌f1 (t) − 􏼒Al f1 􏼓(s)􏼌􏼌􏼌􏼌
1
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
􏼌􏼌
∧
∧
∧
􏼌
􏼌
􏼌
􏼌
􏼌
􏼌
+ 􏼌􏼌􏼌􏼌f2 (t) − 􏼒Al f2 􏼓(s)􏼌􏼌􏼌􏼌cosh􏼌􏼌􏼌􏼌f2 (t) − 􏼒Al f2 􏼓(s)􏼌􏼌􏼌􏼌 − sinh􏼌􏼌􏼌􏼌f2 (t) − 􏼒Al f2 􏼓(s)􏼌􏼌􏼌􏼌􏼓μ(s)μ(t)
r
⎝ 􏽘
≤⎛
⎠,
ω6 (t) f1 (t)cosh f1 (t) − sinh f2 (t) + f2 (t)cosh f2 (t) − sinh f2 (t)􏼁μ(t)⎞
t ∈ [a,∞)T
(91)
􏽥 ⟶ R+ , where
holds for all Δ􏽥μ-integrable functions f1 , f2 : Ξ
􏽢 l fi 􏼑(s) �
􏼐A
1
σ(s) − a
Proof. We get the result from Teorem 4 by taking
m
fi (t)μ(t),
􏽘
i � 1, 2,
(92)
t∈[a,σ(s))T
Φ(s) � 􏽘 si coshsi − sinhsi 􏼁,
(94)
i�1
and
□
with n � 1 and m � 2.
(1/r)
⎝
ω6 (t) � ⎛
􏽘
s ∈ [t,∞)T
ξ(s)
⎠
μ(s)⎞
(σ(s) − a)r
.
(93)
Remark 17. If we take r � 1 in Teorem 7, then we get
􏽢 l f1 􏼑(s) − sinh􏼐A
􏽢 l f1 􏼑(s) + 􏼐A
􏽢 l f2 􏼑(s)cosh􏼐A
􏽢 l f2 􏼑(s) − sinh􏼐A
􏽢 l f2 􏼑(s)􏼑
􏽢 l f1 􏼑(s)cosh􏼐A
ξ(s)μ(s)􏼐􏼐A
􏽘
s∈[a,∞)T
+
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
􏼌
􏼌
􏼌
􏼌
􏼌
ξ(s) 􏼌􏼌􏼌
􏽢 l f1 􏼑(s)􏼌􏼌􏼌􏼌cosh􏼌􏼌􏼌􏼌f1 (t) − 􏼐A
􏽢 l f1 􏼑(s)􏼌􏼌􏼌􏼌 − sinh􏼌􏼌􏼌f1 (t) − A
􏽢 l f1 (s)􏼌􏼌􏼌
􏼒􏼌􏼌f1 (t) − 􏼐A
σ(s) − a
(95)
􏼌􏼌
􏼌
􏼌
􏼌
􏼌
􏼌
􏽢 l f2 (s)􏼌􏼌􏼌cosh􏼌􏼌􏼌f2 (t) − A
􏽢 l f2 (s)􏼌􏼌􏼌 − sinh􏼌􏼌􏼌f2 (t) − A
􏽢 l f2 (s)􏼌􏼌􏼌􏼑μ(s)μ(t)
+ 􏼌􏼌f2 (t) − A
≤
􏽘
ω7 (t) f1 (t)cosh f1 (t) − sinh f1 (t) + f2 (t)cosh f2 (t) − sinh f2 (t)􏼁μ(t),
t∈[a,∞)T
􏽢 l fi )(s), i � 1, 2 is defned by (93) and
where (A
ω7 (t) �
􏽘
s∈[t, ∞)T
ξ(s)
μ(s),
σ(s) − a
which is [10], (Corollary 4.3).
(96)
Theorem 8. Assuming T be an isolated time scale,
􏽥 � [a, ∞)T ,
ξ: Ξ ⟶ R+ is Δμ-integrable function and Ξ � Ξ
a ≥ 0. Ten,
Journal of Mathematics
15
2
r
2
ξ(s)􏼔 Al f1 􏼁 (s)ln Al f1 􏼁(s) + Al f2 􏼁 (s)ln Al f2 􏼁(s)􏼕 μ(s)
􏽘
s∈[a,∞)T
+r
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
r−
ξ(s)
2
2
􏼒 Al f1 􏼁 (s)ln Al f1 􏼁(s) + Al f2 􏼁 (s)ln Al f2 􏼁(s)􏼓
σ(s) − a
1
􏼌􏼌2 􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌2 􏼌􏼌
􏼌􏼌
􏼌􏼌
× 􏼒􏼌􏼌f1 (t) − Al f1 (s)􏼌􏼌 ln􏼌􏼌f1 (t) − Al f1 (s)􏼌􏼌 + 􏼌􏼌f2 (t) − Al f2 (s)􏼌􏼌 ln􏼌􏼌f2 (t) − Al f2 (s)􏼌􏼌􏼓μ(s)μ(t)
(97)
r
⎝ 􏽘
≤⎛
⎠,
ω8 (t)􏼐f21 (t)ln f1 (t) + f22 (t)ln f2 (t)􏼑μ(t)⎞
t ∈ [a,∞)T
􏽥 ⟶ R+ , where
holds for all Δ􏽥μ-integrable functions f1 , f2 : Ξ
Al fi 􏼁(s) �
1
σ(s) − a
Proof. We get the result from Teorem 4 by taking
m
f(t)μ(t)i ,
􏽘
i � 1, 2,
Φ(s) � 􏽘 s2i ln si ,
(98)
t∈[a,σ(s))T
(100)
i�1
and
□
with the assumption 0 ln 0 � 0.
(1/r)
⎝
ω9 (t) � ⎛
􏽘
s ∈ [t,∞)T
ξ(s)
⎠
μ(s)⎞
(σ(s) − a)r
.
(99)
2
Remark 18. If we take r � 1 in Teorem 8, then we get
2
ξ(s)􏼒 Al f1 􏼁 (s)ln Al f1 􏼁(s) + Al f2 􏼁 (s)ln Al f2 􏼁(s)􏼓μ(s)
􏽘
s∈[a,∞)T
􏼌􏼌
􏼌 􏼌
􏼌
􏼌􏼌f1 (t) − Al f1 (s)􏼌􏼌􏼌2 ln􏼌􏼌􏼌f1 (t) − Al f1 (s)􏼌􏼌􏼌
+
≤
ξ(s) ⎜
⎟
⎛
⎞
⎜
⎟
⎜
⎟
⎝
⎠μ(s)μ(t)
􏼌􏼌
􏼌􏼌2 􏼌􏼌
􏼌􏼌
σ(s)
−
a
t∈[a,∞)T s∈[t,∞)T
􏼌
􏼌
􏼌
􏼌
+􏼌f2 (t) − Al f2 (s)􏼌 ln􏼌f2 (t) − Al f2 (s)􏼌
􏽘
􏽘
(101)
ω9 (t)􏼐f21 (t)ln f1 (t) + f22 (t)ln f2 (t)􏼑μ(t),
􏽘
t∈[a,∞)T
where (Al fi )(s), i � 1, 2 is defned by (99) and
ω9 (t) �
􏽘
s∈[t,∞)T
ξ(s)
μ(s),
σ(s) − a
(102)
Theorem 9. Assuming T be an isolated time scale,
􏽥 � [a, ∞)T ,
ξ: Ξ ⟶ R+ is Δμ-integrable function and Ξ � Ξ
a ≥ 0. Ten,
which is [6], (Corollary 4.4).
r
ξ(s)􏽨􏼐ln􏼐1 + 􏼐A€l f1 􏼑(s) + 􏼐A€l f2 􏼑(s)􏼑 − 􏼐A€l f1 􏼑(s) − 􏼐A€l f2 􏼑(s)􏽩 μ(s)
􏽘
s∈[a,∞)T
+r
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
ξ(s)
r−
􏼐ln􏼐1 + 􏼐A€l f1 􏼑(s) + 􏼐A€l f2 􏼑(s)􏼑 − 􏼐A€l f1 􏼑(s) − 􏼐A€l f2 􏼑(s)􏼑
σ(s) − a
1
􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌
× 􏼐ln􏼐1 + 􏼌􏼌f1 (t) − A€l f1 (s)􏼌􏼌 + 􏼌􏼌f2 (t) − A€l f2 (s)􏼌􏼌􏼑 − 􏼌􏼌f1 (t) − A€l f1 (s)􏼌􏼌 − 􏼌􏼌f2 (t) − A€l f2 (s)􏼌􏼌􏼑μ(s)μ(t)
r
⎝
≤⎛
􏽘
t ∈ [a,∞)T
⎠,
ω10 (t)􏼂ln 1 + f1 (t) + f2 (t)􏼁 − f1 (t) − f2 (t)􏼃μ(t)⎞
(103)
16
Journal of Mathematics
holds for all Δ􏽥μ-integrable functions f1 , f2 : Ξ2 ⟶ R+ ,
where
1
􏼐A€l fi 􏼑(s) �
σ(s) − a
f(t)i μ(t),
􏽘
i � 1, 2,
m
m
⎝1 + 􏽘 s ⎞
⎠ − 􏽘s .
Φ(s) � ln⎛
i
i
(104)
i�1
(106)
i�1
□
t∈[a,σ(s))T
and
⎝
ω10 (t) � ⎛
Proof. We get the result from Teorem 4 by taking
􏽘
s ∈ [t,∞)T
ξ(s)
⎠
μ(s)⎞
(σ(s) − a)r
Remark 19. If we take r � 1 in Teorem 9, then we get
(1/r)
.
(105)
ξ(s)􏽨ln􏼐1 + 􏼐A€l f1 􏼑(s) + 􏼐A€l f2 􏼑(s)􏼑 − 􏼐A€l f1 􏼑(s) − 􏼐A€l f2 􏼑(s)􏽩μ(s)
􏽘
s∈[a,∞)T
+
􏽘
􏽘
t∈[a,∞)T s∈[t,∞)T
≤
􏽘
􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌 􏼌􏼌
􏼌􏼌
ξ(s)
􏼐ln(1 + 􏼌􏼌f1 (t) − A€l f1 (s)􏼌􏼌 + 􏼌􏼌f2 (t) − A€l f2 (s)􏼌􏼌􏼑 − 􏼌􏼌f1 (t) − A€l f1 (s)􏼌􏼌 − 􏼌􏼌f2 (t) − A€l f2 (s)􏼌􏼌μ(s)μ(t)
σ(s) − a
ω11 (t)􏼂ln 1 + f1 (t) + f2 (t)􏼁 − f1 (t) − f2 (t)􏼃μ(t),
t∈[a,∞)T
(107)
where (A€l fi )(s), i � 1, 2 is defned by (105) and
ω11 (t) �
􏽘
s∈[t,∞)T
ξ(s)
μ(s),
σ(s) − a
Authors’ Contributions
(108)
which is [10], (Corollary 4.5).
H. M. Rezk, H. A. Abd El-Hamid, R. Butush, and
G. AlNemer performed software analysis and wrote the
original draft. H. M. Rezk and M. Zakarya reviewed and
edited the manuscript. All the authors have read and agreed
to the published version of the manuscript.
6. Conclusion and Future Work
Tis research article is dedicated for some general dynamic
inequalities of Hardy’s type and their converses on time
scales. Tese inequalities are considered rather in general
terms and contain a number of special integral inequalities.
In particular, our fndings can be seen as refnements of
some recent results closely linked to classical Hardy and
Pólya–Knopp inequalities on time scale. We use some algebraic inequalities such as the Minkowski inequality, the
refned Jensen inequality, and the Bernoulli inequality on
time scales to prove the essential results in this paper. Our
computed outcomes can be very useful as a starting point to
get some continuous inequalities as special cases. In the
future, such inequalities can be introduced by using fractional integrals and fractional derivatives of the Riemann− Liouville type on time scales. It will also be very
enjoyable to introduce such inequalities in quantum
calculus.
Data Availability
No data were used to support this study.
Conflicts of Interest
Te authors declare that they have no conficts of interest.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers
Supporting Project number (PNURSP2022R45), Princess
Nourah bint Abdulrahman University, Riyadh, Saudi
Arabia.
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