EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 469-485
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

The Quotient Inequalities

Benedict Barnes1,∗, C. Sebil1, I. K. Dontwi1

1 Department of Mathematics, Kwame Nkrumah University of Science and Technology,
Kumasi, Ghana

Abstract. This paper contributes to the field of inequalities, specifically, the relationships among
the norms of products of elements or vectors or functions and their quotients. Thus, we established
that the norm of product of two vectors or functions is less than or equal to the norm of its quotient
if the norm of the denominator is less than or equal to one. On the other hand, we prove that the
norm of the quotient of two vectors or functions is less than or equal to the norm of their product.
In addition, we introduce the proofs of inequalities including the norm of index power of products
and their quotients, and then applied these inequalities to estabish properties of some functional
spaces, as well as, extention of some of the results in these functional spaces.

2010 Mathematics Subject Classifications: 44B56, 44B57

Key Words and Phrases: Quotient inequalities, first quotient inequality, second quotient
inequality, index power quotient inequalities

1. Introduction

Inequalities play a central role in mathematical analysis with numerous applications
in solving ill-posed differential equations, approximation theory, optimization theory, nu-
merical analysis, probability theory and statistics. The authors in [1] obtained estimates
relating martingale difference sequences in the complex uniformly convex spaces. In [2],
they obtained some estimates for geometric inequalities and compared these inequalities.
The authors in [3] provided an alternative way of proving the quasi-normed linear space
through binomial inequalities.

In this paper, the new inequalities; first and second quotient inequalities, for the con-
tinuous mappings or operators on a real Hilbert space, complex Hilbert space, Banach
space and Hölder’s spaces and Sobolev spaces are provided. These quotient inequalities
are used in obtaining various new inequalities for continuous functions of both selfadjoint
and adjoint operators.

The section one of this paper contains the general overview of the inequalities with
emphasizes on the quotient inequalities. In section 2, the preliminary results including

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3384

Email addresses: ewiekwamina@gmail.com (B. Barnes), bbarnes.cos@knust.edu.gh (B. Barnes)

http://www.ejpam.com 469 c© 2019 EJPAM All rights reserved.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 470

definitions and theorem which are necessary in establishing the quotient inequalities are
provided. The quotient inequalities; the first and second quotient inequalities, are intro-
duced in section 3 of this paper. These inequalities are used to establish the relationships
among the norms of product of functions and their quotient counterparts in Lp spaces,
Hilbert space, Sobolev spaces, Holder’s space, Banach space and unitary space.

2. Some Preliminary Results

In this section, the definitions regarding the quotient inequalities and index power
inequalities are provided.

Definition 1 (First and Second Product Inequalities). Let a1 and a2 be any two positive
real numbers, then

(i)‖a1‖‖a2‖ ≤ ‖a1‖+ ‖a2‖, ∀a1, a2 ∈ [0, 2].

(ii)‖a1‖+ ‖a2‖ ≤ ‖a1‖‖a2‖, ∀a1, a2 ∈ [2,∞).

See [4].

Definition 2 (Product-normed Linear Space). Let U be a linear space over [0, 2] ⊆ R. A
vector space with a product-norm u→ ‖U‖ satisfying real-valued function ‖ · ‖,

‖ · ‖pn : U → [0,∞),

such that for arbitrary u, v ∈ U, α ∈ [0, 2], the following conditions are satisfied:

P1. ‖u‖ ≥ 0, and ‖u‖ = 0, if and only if u = 0

P2. ‖αu‖ = |α|‖u‖, α ∈ [0, 2], and u ∈ U
P3. ‖u+ v‖pn ≤ ‖u‖pn + ‖v‖pn, ∀u, v ∈ U

We call ‖ · ‖pn a product norm, if in addition to P1− P3, the u and v satisfy

P4(a). ‖u‖pn‖y‖pn ≤ ‖u‖pn + ‖v‖pn ∀u, v ∈ [0, 2] (first product inequality)

P4(b).‖u‖+ ‖v‖ ≤ ‖u‖‖v‖, ∀u, v ∈ [2,∞) (second product inequality).

See [5].

Definition 3 (Young’s Inequality). For 1 < p < ∞, q the conjugate of p, and any two
positive numbers a and b, then

ab ≤ 1

p
ap +

1

q
aq, p, q ≤ 1.

See [6].



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 471

Definition 4 (Contractive Mapping). Let T : X → X be a mapping from a complete
normed linear space X into itself. The Lipschitz continuity on T is said to be a contraction
if

‖T (u)− T (v)‖ ≤ λ‖u− v‖, ∀ u, v ∈ X, and 0 < λ < 1.

See [7].

Definition 5. Let γ ∈ (0, 1]. We say a function T : X → Y is Hölder continuous of
exponent γ at xo ∈ X, if

‖u(x)− u(xo)‖ ≤ L‖x− xo‖γ ,

where L is boundedness constant may depend on X, xo, γ and T . See [8].

Theorem 1 (Gagliardo-Nirenberg-Sobolev inequality). Let 1 ≤ p < n. Then there exists
a constant C > 0 (depending on p and n) such that

‖u‖p∗,Rn ≤ C‖∇u‖p,Rn , u ∈W 1,p(Rn).

In particular, we have the continuous imbedding

W 1,p(Rn) ↪→ Lp∗(Rn).

For example, see authors in [9].

Definition 6. Let A and B be selfadjoint operators with Sp(A), Sp(B) ⊆ [m,M ] for some
real numbers m < M . If f : [m,M ] → R is of r − L−Hölder type. Thus, for a given
r ∈ (0, 1] and L > 0, we have∣∣∣f(s)− f(t)

∣∣∣ ≤ L∣∣∣s− t∣∣∣r, ∀ s, t ∈ [m,M ],

then we have the Ostrowski type inequality for selfadjoint operators:∣∣∣f(s)− 〈f(A)x, x〉
∣∣∣ ≤ [1

2
(M −m) + |s− m+M

2
|
]r
, ∀ s ∈ [m,M ] and x ∈ H with ‖x‖ = 1.

Moreover, we have∣∣∣〈f(B)y, y〉 − 〈f(A)x, x〉
∣∣∣ ≤ 〈∣∣∣f(B)− 〈f(A)x, x〉 · 1H

∣∣∣
y
, y
〉
.

≤ L
[1

2
(M −m) + 〈|B − m+M

2
· 1H |y, y〉

]r
,

∀ x, y ∈ H with ‖x‖ = ‖y‖ = 1.

See [10].



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 472

3. Main Result

In this section, we derive both the first and second quotient inequalities in a suitable
functional space.

Theorem 2 (First Quotient Inequality). Suppose that x and y are two real numbers or
two real-valued vectors, then

‖x‖‖y‖ ≤ ‖x‖
‖y‖

, ∀ 0 ≤ ‖x‖ <∞ and 0 < ‖y‖ ≤ 1,

where the equality occurs at either ‖x‖ = 0 or ‖y‖ = 1.

Proof : Setting f(x, y) = −νx2 − 2νx2y2, the function attains its maximum value at
zero, for all (x, y) ∈ R2 and ν ∈ [0, 1]. We can see that:

f(x, y) = −
(
νx2 + 2νx2y2

)
≤ 0

f(x, y) = −
(
νx2 + x2(2νy2)

)
≤ 0

f(x, y) = −
(
νx2 + x2(2νy2)

)
≤ −

(
x2ν + x2y4ν

)
≤ 0

By transitivity law, we obtain

−
(
x2ν + x2y4ν

)
≤ 0

⇒ −x2y4ν ≤ x2ν

⇒ ‖− x2νy2ν‖ = ‖ x2ν

x2(1−ν)y2ν
‖

⇒ ‖x‖2ν‖y‖2ν ≤ ‖x‖2ν

‖x‖2(1−ν)‖y‖2ν
. (1)

Setting ν = 1 in inequality (1) yields

‖x‖‖y‖ ≤ ‖x‖
‖y‖

. (2)

We search for the regions for which ‖x‖ and ‖y‖ hold. Setting ν = 1
4 in inequality (1), we

obtain

‖x‖3 ≤ 1

‖y‖2
. (3)

Again, plugging ν = 1
2 into inequality (1) yields

‖x‖ ≤ 1

‖y‖2
. (4)



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 473

From the inequalities (3) and (4), we consider three situations for which inequality (2)
holds. We observe that

‖x‖3 < ‖x‖
⇒ x(x2 − 1) < 0

⇒ 0 < ‖x‖ < 1. (5)

Also, we can see that the two inequalities in (3) and (4) are equal if,

‖x‖3 = ‖x‖
⇒ x(x2 − 1) = 0

⇒ ‖x‖ = 0, or ‖x‖ = 1. (6)

Lastly, the norms on the left hand sides of inequalities (3) and (4) can be written as:

‖x‖3 > ‖x‖.
⇒ x(x2 − 1) > 0

⇒ ‖x‖ > 0, or ‖x‖ > 1. (7)

The region for which ‖y‖ holds is as follows.

‖y‖2 ≤ 1

‖x‖
⇒ ‖y‖2 ≤ 1, ∀ ‖x‖ = 1

⇒ 0 < ‖y‖ ≤ 1. (8)

Combining the inequalities (5), (7) and (8), and equation (6) together with inequality (2),
we obtain

‖x‖‖y‖ ≤ ‖x‖
‖y‖

, ∀ 0 ≤ ‖x‖ <∞ and 0 < ‖y‖ ≤ 1.

Theorem 3 (Second Quotient Inequality ). Suppose that x and y are any two real numbers
or any two real-valued vectors, then

‖x‖
‖y‖
≤ ‖x‖‖y‖, ∀ ‖x‖ ≥ 1 and ‖y‖ ≥ 1,

where the equality occurs at either ‖x‖ = ‖y‖ = 1 or ‖y‖ = 1.

Proof : We can see that f(x, y) = νx2 + 2νx2y2 attains its minimum value at zero, for
all (x, y) ∈ R2 and ν ∈ [0, 1].

f(x, y) = νx2 + 2νx2y2 ≥ 0



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 474

f(x, y) = νx2 + x2(2νy2) ≥ 0

f(x, y) = νx2 + x2(2νy2) ≥ x2ν + x2y4ν ≥ 0

By transitivity law, we obtain

x2ν + x2y4ν ≥ 0

⇒ −x2ν ≤ x2y4ν

⇒ ‖− x2ν

x2(1−ν)y2ν
‖ = ‖x2νy2ν‖

⇒ ‖x‖2ν

‖x‖2(1−ν)‖y‖2ν
≤ ‖x‖2ν‖y‖2ν . (9)

Setting ν = 1 in inequality (9) yields

‖x‖
‖y‖
≤ ‖x‖‖y‖.

The regions in which the above inequality holds are as follows. Setting ν = 0 in inequality
(9) yields

‖x‖2 ≥ 1

⇒ ‖x‖ ≥ 1.

In order to obtain the region for ‖y‖, we set

f(x, y) = νx2 + (2νx2)y2 ≥ x2ν + x2y4ν ≥ 0

⇒ −x2ν ≤ x4νy2

⇒
∥∥∥ −1

y2(1−ν)

∥∥∥ =
∥∥∥x2νy2ν

∥∥∥
⇒ 1

‖y‖2(1−ν)
≤ ‖x‖2ν‖y‖2ν . (10)

Setting ν = 0 in inequality (10) yields

‖y‖2 ≥ 1

⇒ ‖y‖ ≥ 1.

We observed that the equality occurs at either ‖x‖ = ‖y‖ = 1 or ‖y‖ = 1.

3.1. Illustration of the First Quotient Inequality to the Real Line

In this subsection, the illustration of the first quotient inequality is provided.

Example 1. . Setting a = 2
3 and b = 4

5 , then∥∥∥2

3

∥∥∥∥∥∥4

5

∥∥∥ <
‖2

3‖
‖4

5‖

⇒ 8

15
<

5

6
.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 475

3.2. The Applications of the First Quotient Inequality to LP Spaces

Theorem 4. Suppose that f(x) and g(x) are measurable functions over the domain Ω
such that ∫

Ω
|f(x)|dx ≤ +∞.

and ∫
Ω
|g(x)|dx ≤ 1,

then

‖f‖p‖g‖p ≤
‖f‖p
‖g‖p

.

Proof : We observe that:∣∣∣ ∫
Ω
f(x)g(x)dx

∣∣∣ ≤ ∫
Ω
|f(x)||g(x)|dx.

Applying the first quotient inequality to the term on the right hand side of the above
inequality, we obtain ∣∣∣ ∫

Ω
f(x)g(x)dx

∣∣∣ ≤ ∫
Ω
|f(x)||g(x)|dx ≤

∫
Ω

|f(x)|
|g(x)|

dx

⇒
∫

Ω
|f(x)||g(x)|dx ≤

∫
Ω

|f(x)|
|g(x)|

dx

⇒
∫

Ω
|f(x)||g(x)|dx ≤

∫
Ω |f(x)|dx∫
Ω |g(x)|dx

⇒
(∫

Ω
|f(x)|pdx

) 1
p
(∫

Ω
|g(x)|pdx

) 1
p

=

( ∫
Ω |f(x)|pdx

) 1
p

( ∫
Ω |g(x)|pdx

) 1
p

⇒ ‖f‖p‖g‖p ≤
‖f‖p
‖g‖p

.

3.3. The Applications of the Second Quotient Inequality to LP Spaces

In this subsection, the second quotient inequality is used to estimate the integrals in
Lp spaces.

Theorem 5. Suppose that f(x) and g(x) are measurable functions over the domain Ω
such that ∫

Ω
|f(x)|dx ≥ 1



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 476

and ∫
Ω
|g(x)|dx ≥ 1,

then

‖f‖p
‖g‖p

≤ ‖f‖p‖g‖p.

Proof : We can see that: ∣∣∣ ∫
Ω

f(x)

g(x)
dx
∣∣∣ ≤ ∫

Ω

|f(x)|
|g(x)|

dx∣∣∣ ∫
Ω

f(x)

g(x)
dx
∣∣∣ ≤ ∫

Ω |f(x)|dx∫
Ω |g(x)|dx

Applying the second quotient inequality to the term on the right hand side of the above
inequality yields ∣∣∣ ∫

Ω

f(x)

g(x)
dx
∣∣∣ ≤ ∫

Ω |f(x)|dx∫
Ω |g(x)|dx

≤
∫

Ω
|f(x)||g(x)|dx

⇒
∫

Ω |f(x)|dx∫
Ω |g(x)|dx

≤
∫

Ω
|f(x)||g(x)|dx

⇒

( ∫
Ω |f(x)|pdx

) 1
p

( ∫
Ω |g(x)|pdx

) 1
p

=
(∫

Ω
|f(x)|pdx

) 1
p
(∫

Ω
|g(x)|pdx

) 1
p

⇒ ‖f‖p
‖g‖p

≤ ‖f‖p‖g‖p.

Corollary 1. Suppose that f(x) and g(x) are any two measurable functions defined on R
with ‖f(x)‖ ≤ 1 and ‖g(x)‖ ≤ 1, then∫ b

a

‖f(x)‖
‖g(x)‖

dx ≤
∫ b

a
‖f(x)‖‖g(x)‖dx.

Proof : We observe that:∣∣∣ ∫ b

a

f(x)

g(x)
dx
∣∣∣ ≤ ∫ b

a

∣∣∣f(x)

g(x)

∣∣∣dx
⇒
∣∣∣ ∫ b

a

f(x)

g(x)
dx
∣∣∣ =

∫ b

a

‖f(x)‖
‖g(x)‖

dx ≤
∫ b

a
‖f(x)‖‖g(x)‖dx

⇒
∫ b

a

‖f(x)‖
‖g(x)‖

dx ≤
∫ b

a
‖f(x)‖‖g(x)‖dx.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 477

Corollary 2 (Isotonic linear functional). Let F (T ) be an algebra of real functions defined
on T and L a subclass of F (T ) satisfying the axioms:
(i) f, g ∈ L⇒ f + g ∈ L;
(ii) f ∈ L, α ∈ R⇒ αf ∈ L. A functional A defined on L is an isotonic linear functional
on L provided that:

(a)A(αf + βg) = αA(f) + βA(g), ∀ α, β ∈ R and f, g ∈ L
(b) f(t) ≥ g(t)⇒ A(f) ≥ A(g), t ∈ T.

Proof : To prove the result in corollary 2(a), see for example, author in [11].
We prove the finding in corollary 2(b) by setting ‖f(t)g(t)‖ ≥ 1, then

‖f(t)‖ ≥ 1

‖g(t)‖
.

Applying the second quotient inequality to the right hand side of the above inequality
yields

‖f(t)‖ ≥ 1

‖g(t)‖
≤ ‖g(t)‖

⇒ ‖f(t)‖ ≥ ‖g(t)‖.

This completes the proof.

3.4. Using the quotient Inequalities to obtain the estimate of Hahn-
Banach contraction mapping theorem

In this section, the first product inequality is used to obtain an alternative way for
proving Hahn Banach contraction mapping theorem.

Proposition 1. A contraction mapping T , defined on a complete normed linear space,
has unique fixed point.

Proof : Setting a mapping T : X → X, and let xo ∈ X such that

Txn−1 = xn, ∀ n = 1, 2, . . . .

For any positive integer n, then

‖xm+1 − xm‖ = ‖Txm − Txm−1‖ ≤ α‖xm − xm−1‖
...

‖xm+1 − xm‖ ≤ αm‖x1 − xo‖

‖xm+1 − xm‖ ≤
(
αn+k−1 + αn+k−2 + . . .+ αn

)
‖x1 − xo‖.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 478

where, α, is the boundedness constant. Using the first quotient inequality on the right
hand side of the above inequality yields

‖xm+1 − xm‖ ≤
1(

αn+k−1 + αn+k−2 + . . .+ αn
)‖x1 − xo‖

‖xm+1 − xm‖ =
(1− α)

αn
‖x1 − xo‖.

We can see that (1−α)
αn → 0 as n→∞ for all α ≥ 1. The sequence {Xn}∞n=1 is convergent.

The normed space X is complete since {Xn}∞n=1 has a limit point in X.
Let x be the element of X such that

lim
n→∞

Xn = x.

Thus,

Tx = T ( lim
n→∞

Xn)

Tx = lim
n→∞

TXn.

By the continuity of T . We can see that:

lim
n→∞

Tyn = lim
n→∞

Tyn+1 = y.

Suppose further that Ty1 = y1 and Ty2 = y2. Then

‖y1 − y2‖ = ‖Ty1 − Ty2‖
‖y1 − y2‖ ≤ ‖y1 − y2‖
‖y1 − y2‖ < ‖y1 − y2‖,

which is a contradiction. Thus, the fixed point theorem is unique.

3.5. Using the Product and quotient Inequalities to obtain sharp In-
equalities in Sobolev spaces

Lemma 1. For any 1 ≤ p < n, W 1,p(Rn) ↪→ Lr(Rn) is continuously imbedded, ∀ α ∈
(0, 1] and ‖u‖ ≥ 2.

Proof :Setting α ∈ (0, 1] and u ∈W 1,p(Rn), we see that u ∈ Lp∗(Rn). Then

‖u‖rr ≤ α

∫
Rn

(|u|r + |u|r)dx

‖u‖rr = α
(∫

Rn

|u|rdx+

∫
Rn

|u|rdx
)

‖u‖rr ≤ α
(

(

∫
Rn

|u|pdx)
r
p + (

∫
Rn

|u|p∗dx)
r
p∗
)



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 479

‖u‖r ≤ α
(
‖u‖p + ‖u‖p∗

)
.

Using the second product inequality, we obtain

‖u‖r ≤ α
(
‖u‖p‖u‖p∗

)
⇒ ‖u‖r ≤ α

(
c‖u‖p‖∇u‖p∗

)
⇒ ‖u‖r ≤ αc‖u‖21,p.

This completes the proof.

4. The Applications of the First Quotient Inequality to Unitary Space

In this section, the estimates involving the quotients of norms in the unitary space are
obtained by using both the first and second product inequalities.

Definition 7. Setting

HE
p (ε) = inf

{
sup

( 1

2π

∫ 2π

0
‖x+ eiθy‖pdθ

) 1
p − 1 : ‖y‖ = 1, ‖x‖ = ε

}
, 0 < p <∞, and ε ≥ 1.

Theorem 6. Suppose that (X, ‖ · ‖p) is a continuously quasi-normed space. Then for any
x ∈ X, there exists δ > 0 such that

1

2π

∫ π

0
‖x+ reiθy‖dθ ≤ 1

2π

∫ π

0

‖x‖
‖reiθy‖

dθ.

Proof : Setting 0 < r ≤ 1, we have( 1

2π

∫ π

0
‖x+ reiθy‖pdθ

) 1
p ≤

( 1

2π

∫ π

0
(‖x‖+ ‖reiθy‖)pdθ

) 1
p

⇒
( 1

2π

∫ π

0
(‖x+ reiθy‖)pdθ

) 1
p ≤

( 1

2π

∫ π

0
(‖x‖p + ‖r‖p)dθ

) 1
p ≤

( 1

2π

∫ π

0

‖x‖p

‖r‖p
dθ
) 1

p

⇒
( 1

2π

∫ π

0
‖x‖pdθ

) 1
p ≤

( 1

2π

∫ π

0

‖x‖p

‖r‖p
dθ
) 1

p

⇒ ‖x‖p ≤
‖x‖p
‖r‖p

.

Hence,

1

2π

∫ π

0
‖x+ reiθy‖dθ ≤ 1

2π

∫ π

0

‖x‖
‖reiθy‖

dθ.

The converse inequality of theorem (6) is obvious. Thus, define



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 480

Definition 8.

HE
p (ε) = inf

{
sup

( 1

2π

∫ 2π

0
‖x+ eiθy‖pdθ

) 1
p − 1 : ‖y‖ = 1, ‖x‖ = ε

}
, 0 < p <∞, and ε ≤ 1,

and provide the converse result in corollary 3 below.

Corollary 3. Suppose that (X, ‖ · ‖q) is a continuously quasi-normed space. Then there
exists 0 < p <∞, such that whenever x and y are in X with δ = δ(x, y) > 0, then

‖x‖ ≤
( 1

2π

∫ 2π

0

∥∥∥ x

reiθy

∥∥∥p) 1
p
, 0 < r ≤ δ ≤ 1.

Proof : We see that:( 1

2π

∫ 2π

0

∥∥∥ x

reiθy

∥∥∥pdθ) 1
p

=
( 1

2π

∫ 2π

0

‖x‖p

‖reiθy‖p
dθ
) 1

p
.

Using the first quotient inequality on the right hand side of the above equation, we
obtain

⇒
( 1

2π

∫ 2π

0

∥∥∥ x

reiθy

∥∥∥pdθ) 1
p ≥

(∫ 2π

0

‖x‖p

‖r‖p
) 1

p ≥
(∫ 2π

0
‖x‖p‖r‖p

) 1
p

⇒
(∫ 2π

0
‖x‖p‖r‖p

) 1
p ≤

(∫ 2π

0

‖x‖p

‖r‖p
) 1

p

⇒
(∫ 2π

0
‖x‖p

) 1
p ≤

(∫ 2π

0

‖x‖p

‖r‖p
) 1

p

⇒ ‖x‖p ≤
‖x‖p
‖r‖p

.

It follows that,

‖x‖p ≤
(∫ 2π

0

‖x‖p

‖r‖p
) 1

p
.

Hence,

‖x‖ ≤
( 1

2π

∫ 2π

0

∥∥∥ x

reiθy

∥∥∥p) 1
p
, 0 < r ≤ δ ≤ 1.

Proposition 2. Suppose that (X, ‖ · ‖q) is a continuously quasi-normed space. The fol-
lowing statements are equivalent:
(i) (X, ‖ · ‖) is locally PL-convex;
(ii) there exists 0 < p < ∞, such that whenever x and y are in X with δ = δ(x, y) > 0
such that

‖x‖ ≤
( 1

2π

∫ 2π

0

∥∥∥ x

reiθy

∥∥∥p) 1
p
, 0 < r ≤ δ ≤ 1;

(iii) ln ‖x‖ is a pluri-subharmonic function on X.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 481

5. Quotient Inequalities involving Powers

In this section, the inequalities involving the relationships among index product of
numbers and its quotient of numbers as bases are established for given regions of validity.
Thus, we introduce in this paper the index power quotient inequalities; the first and second
index power quotient inequalities. The main result is expressed in theorem (7), which is
the fundamental tool for proving other results in this paper. involving propositions and
corollaries.

Theorem 7 (First index power quotient inequality). Let T : X → Y be a Banach space.
Then

‖x‖pq ≤ ‖x‖
p
q , ∀ x ∈ X, 0 ≤ p <∞, and 0 < q ≤ 1,

where equality occurs at either p = 0 or q = 1.
Proof : Setting 0 ≤ p <∞, and 0 < q ≤ 1. We can see that:

f(x) = −(pqx+
p

q
x) ≤ 0

⇒ f(x) = −(pqx+
p

q
x) = −(xpq + x

p
q ) ≤ 0

⇒ −(xpq + x
p
q ) ≤ 0

⇒ −xpq ≤ −x
p
q

⇒ ‖x‖pq ≤ ‖x‖
p
q , ∀ x ∈ X.

This completes the proof.
The converse of theorem 7 is stated in theorem 8 below.

Theorem 8 (Second index power quotient inequality). Let T : X → Y be a Banach space.
Then

‖x‖
p
q ≤ ‖x‖pq, ∀ x ∈ X, 1 ≤ p <∞, and 1 ≤ q <∞.

Proof : The proof of theorem 8 is similar to theorem 7.

Theorem 9. Let T : X → Y be a Banach space. Then

‖a‖‖b‖ ≤ pq2‖a‖p + p2q‖b‖q, ∀ a, b ∈ X p, q ≤ 1.

Proof : Using the Young’s inequality, we have

‖ab‖ ≤ ‖qa
p + pbq

pq
‖

‖a‖‖b‖ ≤ ‖qap + pbq‖
‖pq‖

.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 482

Applying the second quotient inequality on the right hand side of the above inequality
yields

‖a‖‖b‖ ≤
(
‖qap + pbq‖

)
‖pq‖

‖a‖‖b‖ ≤ pq2‖a‖p + p2q‖b‖q, p, q ≥ 1.

Theorem 10. Let T : X → Y is a Banach space. Then

pq2‖a‖p + p2q‖b‖q ≤ ‖a‖‖b‖,∀ a, b ∈ X p, q ∈ [0, 1).

Proof : We see that:

‖ab‖ ≤ ‖qa
p + pbq

pq
‖

⇒ ‖a‖‖b‖ ≤ ‖qap + pbq‖
‖pq‖

.

Applying the first quotient inequality on the right hand side of the above inequality yields(
‖qap + pbq‖

)
‖pq‖ ≤ ‖a‖‖b‖

⇒ pq2‖a‖p + p2q‖b‖q ≤ ‖a‖‖b‖, p, q ∈ [0, 1).

6. The Applications of the Index Power Quotient Inequalities to Hölder
Spaces

In this section, the quotient inequalities involving index are introduced.

Theorem 11. Let 0 < p < +∞ and 0 < q < 1 such that |pq | < 1. Then a mapping

T : X → Y is Hölder-type continuous of exponent p
q at xo, if

‖T (x)− T (xo)‖ ≤ L‖x− xo‖
p
q , ∀ x, xo ∈ X,

where L is the p
q−th Hölder coefficient of T .

Proof : By the Hölder continuity of T of exponent γ, we have

‖T (x)− T (xo)‖ ≤ L‖x− xo‖γ , ∀ x, xo ∈ X and γ ∈ (0, 1].

Setting γ = |p||q| and applying the power quotient inequality, we obtain

‖T (x)− T (xo)‖ ≤ L‖x− xo‖pq ≤ L‖x− xo‖
p
q

‖T (x)− T (xo)‖ ≤ L‖x− xo‖
p
q ∀ x, xo ∈ X,

∣∣∣p
q

∣∣∣ ≤ 1.



B. Barnes, C. Sebil, I. K. Dontwi / Eur. J. Pure Appl. Math, 12 (2) (2019), 469-485 483

Theorem 12. Let A and B be selfadjoint operators with Sp(A), Sp(B) ⊆ [m,M ] for some
real numbers m < M . If f : [m,M ] → R is of p

q − L−Hölder type. Thus, for a given
|p||q| ≤ 1 and L > 0, we have:∣∣∣f(s)− f(t)

∣∣∣ ≤ L∣∣∣s− t∣∣∣ pq , ∀ s, t ∈ [m,M ].

Then the Ostrowski type inequality for selfadjoint operators becomes:∣∣∣f(s)− 〈f(A)x, x〉
∣∣∣ ≤ [1

2
(M −m) + |s− m+M

2
|
] p

q
, ∀ s ∈ [m,M ] and x ∈ H with ‖x‖ = 1.

Moreover, we have:∣∣∣〈f(B)y, y〉 − 〈f(A)x, x〉
∣∣∣ ≤ 〈∣∣∣f(B)− 〈f(A)x, x〉 · 1H

∣∣∣
y
, y
〉
.

≤ L
[1

2
(M −m) + 〈|B − m+M

2
· 1H |y, y〉

] p
q
,

∀ x, y ∈ H with ‖x‖ = ‖y‖ = 1,

and H denotes Hilbert space.

Proof : We can see that, using the Ostrowski-type inequality for the Riemann-Stieltjes
integral, we have:

∣∣∣f(s)[u(b)− u(a)]−
∫ b

a
f(t)du(t)

∣∣∣ ≤ L[1

2
(b− a) + |s− a+ b

2
|
]r b∨

a

(u).

Setting r = |p||q| ≤ 1 and applying the first index power quotient inequality, we obtain

∣∣∣f(s)[u(b)− u(a)]−
∫ b

a
f(t)du(t)

∣∣∣ ≤ L
[1

2
(b− a) + |s− a+ b

2
|
]pq b∨

a

(u)

≤ L
[1

2
(b− a) + |s− a+ b

2

] p
q

b∨
a

(u)

∣∣∣f(s)[u(b)− u(a)]−
∫ b

a
f(t)du(t)

∣∣∣ ≤ L
[1

2
(b− a) + |s− a+ b

2
|
] p

q
b∨
a

(u), s ∈ [a, b],

u is a bounded variation on [a, b] and
∨b
a is the total variation of u on [a, b]. Then the

functional f(t) is of p
q − L−Hölder-type on [a, b].

Also, setting u(λ) = gx(λ) = 〈Eλx, x〉, where x ∈ H with ‖x‖ = 1, then

∣∣∣f(s)−
∫ M

m
f(λ)d(〈Eλx, x〉)

∣∣∣ ≤ L
[1

2
(M −m) + |s− m+M

2
|
]pq M∨

m

(g(x))



REFERENCES 484

≤ L
[1

2
(M −m) + |s− m+M

2
|
] p

q
M∨
m

(g(x))

∣∣∣f(s)−
∫ M

m
f(λ)d(〈Eλx, x〉)

∣∣∣ ≤ L
[1

2
(M −m) + |s− m+M

2
|
] p

q
M∨
m

(g(x)).

Again, we see that:〈
|f(B)− 〈f(A)x, x〉 · 1H |y, y

〉
≤

〈[1

2
(M −m) + |B − m+M

2
· 1H |

]pq
y, y
〉

≤
〈[1

2
(M −m) + |B − m+M

2
· 1H |

] p
q
y, y
〉
,

∀ x, y ∈ H, ‖x‖ = ‖y‖ = 1.

This completes the proof.

7. Conclusion

In a nutshell, the first and second quotient inequalities are introduced which extend
some findings in functional spaces such as Lp space, Hilbert space, unitary space, Hölder
spaces and Sobolev spaces. The relationship between the inequality involving norm of
quotient of the vectors or functions and its product counterpart has not been observed over
the years and vice versa. These inequalities do not only establish the relationship between
two mathematical structures but also, useful for establishing many functional properties
such as continuity, boundedness, expansive of the operators in functional spaces.

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[2] M. Agueh, N. Ghoussoub and X. Kang, Geometric inequalities via a general compar-
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[3] B. Barnes, C. Sebil and E. Harris, A note on the quasi-normed linear space, Advances
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[4] B. Barnes, E. D. J. Owusu-Ansah, S. K. Amponsah and C. Sebil, The proofs of
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[5] B. Barnes, I. A. Adjei and C. Sebil and E. Harris, product-normed linear spaces,
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