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Coordination of Classical and Dynamic Inequalities Complying on Time Scales

Muhammad Jibril Shahab Sahir
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

Correspondence: jibrielshahab@gmail.com

Abstract. In this research article, we present extensions of some classical inequalities such asSchweitzer, Pólya–Szegö, Kantorovich and Greub–Rheinboldt inequalities of fractional calculus ontime scales. To investigate generalizations of such types of classical inequalities, we use the timescales Riemann–Liouville type fractional integrals. We explore dynamic inequalities on delta calcu-lus and their symmetric nabla versions. A time scale is an arbitrary nonempty closed subset of thereal numbers. The theory of time scales is applied to combine results in one comprehensive form.The calculus of time scales unifies and extends continuous versions and their discrete and quantumanalogues. By using the calculus of time scales, results are presented in more general form. Thishybrid theory is also widely applied on dynamic inequalities.

1. Introduction
The calculus of time scales was initiated by Stefan Hilger [11]. The three most popular examplesof calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e.,when T = R, T = N and T = qN0 = {qt : t ∈N0} where q > 1. The time scales calculus is studiedas delta calculus, nabla calculus and diamond-α calculus. During the last two decades, manyresearchers investigated several dynamic inequalities [1–4, 7, 16–18]. The basic work on dynamicinequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, DonalO’Regan, Samir Saker and many other authors.There have been recent achievements of the theory and applications of dynamic inequalities ontime scales. From the theoretical point of view, the study provides a harmonious reconciliation andextension of commonly known differential, difference and quantum equations. Moreover, it is animportant tool in many computational, biological, economical and numerical applications.In this paper, it is assumed that all considerable integrals exist and are finite and T is a timescale, a,b ∈T with a < b and an interval [a,b]T means the intersection of a real interval with thegiven time scale.
Received: 22 Aug 2021.
Key words and phrases. time scales; fractional Riemann–Liouville integral; Schweitzer, Pólya–Szegö, Kantorovichand Greub–Rheinboldt inequalities. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 2
2. Preliminaries

We need here basic concepts of delta calculus. The results of delta calculus are adopted frommonographs [7, 8].For t ∈T, the forward jump operator σ : T→T is defined by
σ(t) := inf{s ∈T : s > t}.

The mapping µ : T → R+0 = [0, +∞) such that µ(t) := σ(t) − t is called the forward graininessfunction. The backward jump operator ρ : T→T is defined by
ρ(t) := sup{s ∈T : s < t}.

The mapping ν : T→R+0 = [0, +∞) such that ν(t) := t −ρ(t) is called the backward graininessfunction. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if t < supT and σ(t) = t, then t is called right-dense, and if t > inf T and
ρ(t) = t, then t is called left-dense. If T has a left-scattered maximum M, then Tk = T−{M},otherwise Tk = T.For a function f : T→R, the delta derivative f ∆ is defined as follows:Let t ∈ Tk . If there exists f ∆(t) ∈ R such that for all � > 0, there is a neighborhood U of t,such that

|f (σ(t)) − f (s) − f ∆(t)(σ(t) − s)| ≤ �|σ(t) − s|,

for all s ∈ U, then f is said to be delta differentiable at t, and f ∆(t) is called the delta derivativeof f at t.A function f : T→R is said to be right-dense continuous (rd-continuous), if it is continuous ateach right-dense point and there exists a finite left-sided limit at every left-dense point. The setof all rd-continuous functions is denoted by Crd(T,R).The next definition is given in [7, 8].
Definition 2.1. A function F : T → R is called a delta antiderivative of f : T → R, provided that
F ∆(t) = f (t) holds for all t ∈Tk . Then the delta integral of f is defined by∫ b

a

f (t)∆t = F (b) −F (a).

The following results of nabla calculus are taken from [6–8].If T has a right-scattered minimum m, then Tk = T−{m}, otherwise Tk = T. A function
f : Tk → R is called nabla differentiable at t ∈ Tk , with nabla derivative f∇(t), if there exists
f∇(t) ∈R such that given any � > 0, there is a neighborhood V of t, such that

|f (ρ(t)) − f (s) − f∇(t)(ρ(t) − s)| ≤ �|ρ(t) − s|,

for all s ∈ V .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 3
A function f : T→R is said to be left-dense continuous (ld-continuous), provided it is continuousat all left-dense points in T and its right-sided limits exist (finite) at all right-dense points in T.The set of all ld-continuous functions is denoted by Cld(T,R).The next definition is given in [6–8].

Definition 2.2. A function G : T→R is called a nabla antiderivative of g : T→R, provided that
G∇(t) = g(t) holds for all t ∈Tk . Then the nabla integral of g is defined by∫ b

a

g(t)∇t = G(b) −G(a).

The following definition is taken from [2, 4].
Definition 2.3. For α ≥ 1, the time scale ∆-Riemann–Liouville type fractional integral for a function
f ∈ Crd is defined by

Iαa f (t) =
∫ t
a

hα−1(t,σ(τ))f (τ)∆τ, (1)
which is an integral on [a,t)T, see [9] and hα : T × T → R, α ≥ 0 are the coordinate wiserd-continuous functions, such that h0(t,s) = 1,

hα+1(t,s) =

∫ t
s

hα(τ,s)∆τ, ∀s,t ∈T. (2)
Notice that

I1af (t) =
∫ t
a

f (τ)∆τ,

which is absolutely continuous in t ∈ [a,b]T, see [9].
The following definition is taken from [3, 4].

Definition 2.4. For α ≥ 1, the time scale ∇-Riemann–Liouville type fractional integral for a function
f ∈ Cld is defined by

Jαa f (t) =
∫ t
a

ĥα−1(t,ρ(τ))f (τ)∇τ, (3)
which is an integral on (a,t]T, see [9] and ĥα : T × T → R, α ≥ 0 are the coordinate wiseld-continuous functions, such that ĥ0(t,s) = 1,

ĥα+1(t,s) =

∫ t
s

ĥα(τ,s)∇τ, ∀s,t ∈T. (4)
Notice that

J 1a f (t) =
∫ t
a

f (τ)∇τ,

which is absolutely continuous in t ∈ [a,b]T, see [9].

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 4
We will generalize the following classical inequalities [13] by using the calculus of time scales.First we consider the inequality given by Schweitzer [19] such that(

1

p

p∑
k=1

xk

)(
1

p

p∑
k=1

1

xk

)
≤

(M + m)2

4Mm
, (5)

where 0 < m ≤ xk ≤ M for k = 1, . . . ,p.In the same paper, Schweitzer has also shown that if functions y 7→ f (y) and y 7→ 1
f (y)

areintegrable on [a,b] and 0 < m ≤ f (y) ≤ M on [a,b], then∫ b
a

f (y)dy

∫ b
a

1

f (y)
dy ≤

(M + m)2

4Mm
(b−a)2. (6)

Pólya and Szegö [15] proved that(
p∑
k=1

x2k

)(
p∑
k=1

y2k

)
(

p∑
k=1

xkyk

)2 ≤

√

MN
mn

+
√

mn
MN

2

2 , (7)
where 0 < m ≤ xk ≤ M and 0 < n ≤ yk ≤ N for k = 1, . . . ,p.Kantorovich [12] proved that(

p∑
k=1

xky
2
k

)(
p∑
k=1

1

xk
y2k

)
≤

1

4

(√
M

m
+

√
m

M

)2 ( p∑
k=1

y2k

)2
, (8)

where 0 < m ≤ xk ≤ M and yk ∈ R for k = 1, . . . ,p, and he pointed out that inequality (8) is aparticular case of (7).Greub and Rheinboldt [10] proved that(
p∑
k=1

x2kz
2
k

)(
p∑
k=1

y2k z
2
k

)
≤

(MN + mn)
2

4MNmn

(
p∑
k=1

xkykz
2
k

)2
, (9)

where 0 < m ≤ xk ≤ M, 0 < n ≤ yk ≤ N and zk ∈R for k = 1, . . . ,p with p∑
k=1

z2k < ∞.
3. Main Results

In order to present our main results, first we give a simple proof for an extension of Pólya–Szegö’sinequality by using the time scale ∆-Riemann–Liouville type fractional integral.
Theorem 3.1. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions. Assume that there
exist four positive ∆-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 5
Let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. Then we have the following inequality

Iαa ((f1f2)(x)|w(x)|)I
β
a ((g1g2)(x)|w(x)|)Iαa

(
|w(x)||f (x)|2

)
Iβa
(
|w(x)||g(x)|2

){
Iαa (f1(x)|(wf )(x)|)I

β
a (g1(x)|(wg)(x)|) + Iαa (f2(x)|(wf )(x)|)I

β
a (g2(x)|(wg)(x)|)

}2 ≤ 14.
(10)

Proof. Using the given conditions, for y,z ∈ [a,x]T, ∀x ∈ [a,b]T, we have(
f2(y)

g1(z)
−
|f (y)|
|g(z)|

)
≥ 0,

and (
|f (y)|
|g(z)|

−
f1(y)

g2(z)

)
≥ 0,

which imply that (
f1(y)

g2(z)
+
f2(y)

g1(z)

)
|f (y)|
|g(z)|

≥
|f (y)|2

|g(z)|2
+
f1(y)f2(y)

g1(z)g2(z)
.

Multiplying both sides by g1(z)g2(z)|g(z)|2, we have
f1(y)g1(z)|f (y)g(z)| + f2(y)g2(z)|f (y)g(z)| ≥ g1(z)g2(z)|f (y)|2 + f1(y)f2(y)|g(z)|2. (11)

Multiplying both sides of (11) by hα−1(x,σ(y))|w(y)|hβ−1(x,σ(z))|w(z)| and double integratingover y and z from a to x, respectively, we have
Iαa (f1(x)|w(x)f (x)|)I

β
a (g1(x)|w(x)g(x)|) + I

α
a (f2(x)|w(x)f (x)|)I

β
a (g2(x)|w(x)g(x)|)

≥Iαa
(
|w(x)||f (x)|2

)
Iβa (g1(x)g2(x)|w(x)|) + I

α
a (f1(x)f2(x)|w(x)|)I

β
a

(
|w(x)||g(x)|2

)
. (12)

Applying the AM-GM inequality √ζη ≤ ζ+η
2

, ζ ≥ 0, η ≥ 0, the inequality (12) takes the form
Iαa (f1(x)|w(x)f (x)|)I

β
a (g1(x)|w(x)g(x)|) + I

α
a (f2(x)|w(x)f (x)|)I

β
a (g2(x)|w(x)g(x)|)

≥ 2
√
Iαa (|w(x)||f (x)|2)I

β
a (g1(x)g2(x)|w(x)|)Iαa (f1(x)f2(x)|w(x)|)I

β
a (|w(x)||g(x)|2). (13)

Inequality (13) directly yields inequality (10). The proof of Theorem 3.1 is completed. �
Now, we give an extension of Pólya–Szegö’s inequality by using the time scale ∇-Riemann–Liouville type fractional integral.

Theorem 3.2. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions. Assume that there
exist four positive ∇-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

Let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. Then we have the following inequality

Jαa ((f1f2)(x)|w(x)|)J
β
a ((g1g2)(x)|w(x)|)Jαa

(
|w(x)||f (x)|2

)
Jβa
(
|w(x)||g(x)|2

){
Jαa (f1(x)|(wf )(x)|)J

β
a (g1(x)|(wg)(x)|) + Jαa (f2(x)|(wf )(x)|)J

β
a (g2(x)|(wg)(x)|)

}2 ≤ 14.
(14)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 6
Proof. Similar to the proof of Theorem 3.1. �
Corollary 3.3. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions such that 0 < m ≤
|f (y)| ≤ M < ∞ and 0 < n ≤ |g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α,β ≥ 1 and
hα−1(., .),hβ−1(., .) > 0. Then we have the following inequality

Iαa (|w(x)|)I
β
a (|w(x)|)Iαa

(
|w(x)||f (x)|2

)
Iβa
(
|w(x)||g(x)|2

){
Iαa (|(wf )(x)|)I

β
a (|(wg)(x)|)

}2 ≤ 14
(√

MN

mn
+

√
mn

MN

)2
. (15)

Proof. Putting f1 = m, f2 = M, g1 = n and g2 = N in Theorem 3.1, we get the inequality (15). �
Corollary 3.4. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions such that 0 < m ≤
|f (y)| ≤ M < ∞ and 0 < n ≤ |g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α,β ≥ 1 and
ĥα−1(., .), ĥβ−1(., .) > 0. Then we have the following inequality

Jαa (|w(x)|)J
β
a (|w(x)|)Jαa

(
|w(x)||f (x)|2

)
Jβa
(
|w(x)||g(x)|2

){
Jαa (|(wf )(x)|)J

β
a (|(wg)(x)|)

}2 ≤ 14
(√

MN

mn
+

√
mn

MN

)2
.

(16)
Proof. Similar to the proof of Corollary 3.3. �
Remark 3.1. Let T = R, α,β > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. Then (10) reduces to

Iα0 ((f1f2)(x))I
β
0 ((g1g2)(x))I

α
0

(
f 2(x)

)
Iβ0
(
g2(x)

){
Iα0 ((f1f )(x))I

β
0 ((g1g)(x)) + I

α
0 ((f2f )(x))I

β
0 ((g2g)(x))

}2 ≤ 14, (17)
as given in [14].
Theorem 3.5. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions. Assume that there
exist four positive ∆-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

Let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. Then we have the following inequality

Iαa
(
|w(x)||f (x)|2

)
Iβa
(
|w(x)||g(x)|2

)
≤Iαa

(
f2(x)

g1(x)
|(wf g)(x)|

)
Iβa
(
g2(x)

f1(x)
|(wf g)(x)|

)
. (18)

Proof. Using the given condition, for y ∈ [a,x]T, ∀x ∈ [a,b]T, we have
|f (y)|2 ≤

f2(y)

g1(y)
|f (y)g(y)|.

Multiplying both sides of the last inequality by hα−1(x,σ(y))|w(y)| and integrating over y from
a to x, we have∫ x

a

hα−1(x,σ(y))|w(y)||f (y)|2∆y ≤
∫ x
a

hα−1(x,σ(y))
f2(y)

g1(y)
|w(y)||f (y)g(y)|∆y. (19)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 7
The inequality (19) takes the form

Iαa
(
|w(x)||f (x)|2

)
≤Iαa

(
f2(x)

g1(x)
|w(x)||f (x)g(x)|

)
. (20)

Similarly, we have that
Iβa
(
|w(x)||g(x)|2

)
≤Iβa

(
g2(x)

f1(x)
|w(x)||f (x)g(x)|

)
. (21)

Multiplying (20) and (21), we get the desired inequality (18). �
Theorem 3.6. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions. Assume that there
exist four positive ∇-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

Let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. Then we have the following inequality

Jαa
(
|w(x)||f (x)|2

)
Jβa
(
|w(x)||g(x)|2

)
≤Jαa

(
f2(x)

g1(x)
|(wf g)(x)|

)
Jβa
(
g2(x)

f1(x)
|(wf g)(x)|

)
.(22)

Proof. Similar to the proof of Theorem 3.5. �
Corollary 3.7. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions. Assume that there
exist four positive constants m, M, n and N such that 0 < m ≤ |f (y)| ≤ M < ∞ and 0 < n ≤
|g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α,β ≥ 1 and hα−1(., .),hβ−1(., .) > 0. Then
we have the following inequality

Iαa
(
|w(x)||f (x)|2

)
Iβa
(
|w(x)||g(x)|2

)
Iαa (|(wf g)(x)|)I

β
a (|(wf g)(x)|)

≤
MN

mn
. (23)

Proof. Putting f1 = m, f2 = M, g1 = n and g2 = N in Theorem 3.5, we get the desired inequality.
�

Corollary 3.8. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions. Assume that there
exist four positive constants m, M, n and N such that 0 < m ≤ |f (y)| ≤ M < ∞ and 0 < n ≤
|g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α,β ≥ 1 and ĥα−1(., .), ĥβ−1(., .) > 0. Then
we have the following inequality

Jαa
(
|w(x)||f (x)|2

)
Jβa
(
|w(x)||g(x)|2

)
Jαa (|(wf g)(x)|)J

β
a (|(wf g)(x)|)

≤
MN

mn
. (24)

Proof. Similar to the proof of Corollary 3.7. �
Remark 3.2. Let T = R, α > 0, α = β, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. Then (23) reducesto

Iα0
(
f 2(x)

)
Iα0
(
g2(x)

){
Iα0 (f (x)g(x))

}2 ≤ MNmn . (25)Inequality (25) may be found in [5].

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 8
Theorem 3.9. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions. Assume that there
exist four positive ∆-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

Let α ≥ 1 and hα−1(., .) > 0. Then we have the following inequality

Iαa
(
g1(x)g2(x)|w(x)||f (x)|2

)
Iαa
(
f1(x)f2(x)|w(x)||g(x)|2

)
{Iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}

2
≤

1

4
. (26)

Proof. Using the given conditions, for y ∈ [a,x]T, ∀x ∈ [a,b]T, we have(
f2(y)

g1(y)
−
|f (y)|
|g(y)|

)
≥ 0,

and (
|f (y)|
|g(y)|

−
f1(y)

g2(y)

)
≥ 0.

Multiplying the last two inequalities, we have(
f2(y)

g1(y)
−
|f (y)|
|g(y)|

)(
|f (y)|
|g(y)|

−
f1(y)

g2(y)

)
≥ 0,

which implies (
f1(y)

g2(y)
+
f2(y)

g1(y)

)
|f (y)|
|g(y)|

≥
|f (y)|2

|g(y)|2
+
f1(y)f2(y)

g1(y)g2(y)
.

Multiplying both sides by g1(y)g2(y)|g(y)|2, we have
f1(y)g1(y)|f (y)g(y)| + f2(y)g2(y)|f (y)g(y)| ≥ g1(y)g2(y)|f (y)|2 + f1(y)f2(y)|g(y)|2. (27)

Multiplying both sides of (27) by hα−1(x,σ(y))|w(y)| and integrating over y from a to x, we have
Iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)

≥Iαa
(
g1(x)g2(x)|w(x)||f (x)|2

)
+ Iαa

(
f1(x)f2(x)|w(x)||g(x)|2

)
. (28)

Applying the AM-GM inequality, we get
Iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)

≥ 2
√
Iαa (g1(x)g2(x)|w(x)||f (x)|2)Iαa (f1(x)f2(x)|w(x)||g(x)|2). (29)

Analogously, we have that
Iαa
(
g1(x)g2(x)|w(x)||f (x)|2

)
Iαa
(
f1(x)f2(x)|w(x)||g(x)|2

)
≤

1

4
{Iαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}

2
. (30)

This directly yields the desired inequality (26). �

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 9
Theorem 3.10. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions. Assume that there
exist four positive ∇-integrable functions f1, f2, g1 and g2 such that:

0 < f1(y) ≤ |f (y)| ≤ f2(y) < ∞and 0 < g1(y) ≤ |g(y)| ≤ g2(y) < ∞, (y ∈ [a,x]T,∀x ∈ [a,b]T).

Let α ≥ 1 and ĥα−1(., .) > 0. Then we have the following inequality

Jαa
(
g1(x)g2(x)|w(x)||f (x)|2

)
Jαa

(
f1(x)f2(x)|w(x)||g(x)|2

)
{Jαa ((f1(x)g1(x) + f2(x)g2(x)) |w(x)||f (x)g(x)|)}

2
≤

1

4
. (31)

Proof. Similar to the proof of Theorem 3.9. �
Remark 3.3. Let T = R, α > 0, a = 0, x > 0, w ≡ 1, f > 0 and g > 0. Then (26) reduces to

Iα0
(
g1(x)g2(x)f

2(x)
)
Iα0
(
f1(x)f2(x)g

2(x)
){

Iα0 ((f1(x)g1(x) + f2(x)g2(x)) f (x)g(x))
}2 ≤ 14. (32)

Inequality (32) may be found in [14].
Corollary 3.11. Let w,f,g ∈ Crd ([a,b]T,R−{0}) be ∆-integrable functions. Assume that there
exist four positive constants m, M, n and N such that 0 < m ≤ |f (y)| ≤ M < ∞ and 0 < n ≤
|g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α ≥ 1 and hα−1(., .) > 0. Then we have the
following inequality

Iαa
(
|w(x)||f (x)|2

)
Iαa
(
|w(x)||g(x)|2

)
{Iαa (|w(x)||f (x)g(x)|)}

2
≤

1

4

(√
MN

mn
+

√
mn

MN

)2
. (33)

Proof. Putting f1 = m, f2 = M, g1 = n and g2 = N in Theorem 3.9, we get the desired inequality(33). �
Corollary 3.12. Let w,f,g ∈ Cld ([a,b]T,R−{0}) be ∇-integrable functions. Assume that there
exist four positive constants m, M, n and N such that 0 < m ≤ |f (y)| ≤ M < ∞ and 0 < n ≤
|g(y)| ≤ N < ∞ on the set [a,x]T, ∀x ∈ [a,b]T. Let α ≥ 1 and ĥα−1(., .) > 0. Then we have the
following inequality

Jαa
(
|w(x)||f (x)|2

)
Jαa

(
|w(x)||g(x)|2

)
{Jαa (|w(x)||f (x)g(x)|)}

2
≤

1

4

(√
MN

mn
+

√
mn

MN

)2
. (34)

Proof. Similar to the proof of Corollary 3.11. �
Remark 3.4. We have the following:

(i) Let α = 1, T = Z, a = 1, x = b = p + 1, xk > 0, w(k) = wk = 1xk , f (k) = xk for
k = 1, . . . ,p and n = g = N = 1. Then inequality (33) reduces to inequality (5).(ii) Let α = 1, T = R, x = b, 0 < m ≤ f (y) ≤ M < ∞, w(y) = 1

f (y)
on [a,b] and

n = g = N = 1. Then inequality (33) reduces to inequality (6).(iii) Let α = 1, T = Z, a = 1, x = b = p + 1, w ≡ 1, f (k) = xk > 0 and g(k) = yk > 0 for
k = 1, . . . ,p. Then inequality (33) reduces to inequality (7).

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Eur. J. Math. Anal. 10.28924/ada/ma.3.12 10
(iv) Let α = 1, T = Z, a = 1, x = b = p + 1, xk > 0, yk ∈R, w(k) = wk = 1xk y2k , f (k) = xk for

k = 1, . . . ,p and n = g = N = 1. Then inequality (33) reduces to inequality (8).(v) Let α = 1, T = Z, a = 1, x = b = p + 1, zk ∈ R, w(k) = wk = z2k , f (k) = xk > 0 and
g(k) = yk > 0 for k = 1, . . . ,p. Then inequality (33) reduces to inequality (9).

4. Conclusion
The subject of dynamic inequalities on time scales has become a crucial field of pure and appliedmathematics. Many researchers developed interesting results concerning fractional calculus on timescales. Due to utility of dynamic inequalities in many branches of mathematics, this field is given aprominent importance. This field has a wide scope. Recently, interesting results have obtained byusing Specht’s ratio and Kantorovich’s ratio on time scales as given in [18]. By using these ratios,we can explore further results.Dynamic inequalities may be extended by applying other techniques such as diamond-α inte-gral, which is defined as a linear operator of delta and nabla integrals on time scales. Quantumcalculus, α,β-symmetric quantum calculus, functional generalization, fractional derivatives and n-tuple diamond-alpha integral are some other developed techniques and we will continue them toinvestigate other dynamic inequalities in future research.

References
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https://doi.org/10.1007/978-3-319-11002-8.[2] G.A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comp. Model. 52(2010) 556–566. https://doi.org/10.1016/j.mcm.2010.03.055.[3] G.A. Anastassiou, Foundations of nabla fractional calculus on time scales and inequalities, Comp. Math. Appl. 59(2010) 3750–3762. https://doi.org/10.1016/j.camwa.2010.03.072.[4] G.A. Anastassiou, Integral operator inequalities on time scales, Int. J. Diff. Equ. 7 (2012) 111–137.[5] A. Anber, Z. Dahmani, New integral results using Pólya–Szegö inequality, Acta Comment. Univ. Tart. Math. 17(2013) 171–178. https://doi.org/10.12697/acutm.2013.17.15.[6] D. Anderson, J. Bullock, L. Erbe, A. Peterson, H. Tran, Nabla dynamic equations on time scales, Pan-Amer. Math.J. 13 (2003) 1–47.[7] M. Bohner, A. Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001.[8] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.
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[12] L.V. Kantorovich, Functional analysis and applied mathematics (Russian), Uspehi Mat. Nauk (N.S.). 3 (1948) 89–185(in particular, pp. 142–144) [also translated from Russian into English by C.D. Benster, Nat. Bur. Standards Rep.No. 1509. 1952, 202 pp. (in particular, pp. 106–109)].[13] D.S. Mitrinović, Analytic inequalities, Springer-Verlag, Berlin, 1970. https://doi.org/10.1007/

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https://doi.org/10.1007/978-3-642-99970-3
https://doi.org/10.7153/jmi-10-38
https://doi.org/10.1007/978-3-662-38381-0
https://doi.org/10.1007/978-3-662-38381-0
https://doi.org/10.15826/umj.2018.2.010

	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Conclusion
	References

