International Journal of Analysis and Applications

Volume 17, Number 1 (2019), 14-25

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

DOI: 10.28924/2291-8639-17-2019-14

ON I-ASYMPTOTICALLY LACUNARY STATISTICAL EQUIVALENCE OF
FUNCTIONS ON AMENABLE SEMIGROUPS

ÖMER KİŞİ∗, BURAK ÇAKAL

Department of Mathematics, Faculty of Science, Bartın University, 74100, Bartın, Turkey

∗Corresponding author: okisi@bartin.edu.tr

Abstract. In this study we define the notions of asymptotically paper, we introduce the concept of I-

asymptotically statistical equivalent and I-asymptotically lacunary statistical equivalent functions defined

on discrete countable amenable semigroups. In addition to these definitions, we give some inclusion theorems.

1. Introduction

Fast [5] presented an interesting generalization of the usual sequential limit which he called statistical con-

vergence for number sequences. Schoenberg [24] established some basic properties of statistical convergence

and also studied the concept as a summability method.

Using lacunary sequences Fridy and Orhan defined lacunary statistical convergence in [6]. Also, in another

study, they gave the relationships between the lacunary statistical convergence and the Cesàro summability.

After their definition, Freedman et al. [7] established the connection between the strongly Cesàro summable

sequences and the strongly lacunary summable sequences.

The concept of I-convergence of real sequences is a generalization of statistical convergence which is based

on the structure of the ideal I of subsets of the set of natural numbers. P. Kostyrko et al. [8] introduced the

concept of I-convergence of sequences in a metric space and studied some properties of this convergence.

Received 2018-07-04; accepted 2018-09-07; published 2019-01-04.

2010 Mathematics Subject Classification. 40A05, 40C05.

Key words and phrases. Folner sequence; amenable group; equivalent functions; statistical convergence; lacunary sequences;

I−convergence.
c©2019 Authors retain the copyrights

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

14

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-14


Int. J. Anal. Appl. 17 (1) (2019) 15

Recently, the idea of statistical convergence and lacunary convergence was further extended by Das et al. [2]

to I-statistical convergence, I-lacunary statistical convergence.

In 1993, Mursaleen [15] defined λ-statistical convergence by using the λ sequence. Let Λ denote the set

of all non-decreasing sequences λ = (λn) of positive numbers tending to ∞ such that λn+1 ≤ λn +1 and

λ1 = 1.

Asymptotic equivalence of sequences was introduced by Pobyvanets [18]. Marouf’s work [9] was extension

of Pobyvanets’s work. In 2003, Patterson [16] extended these concepts by presenting an asymptotically

statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability

matrices.

In [17] asymptotically lacunary statistical equivalent which is a natural combination of the definitions

for asymptotically equivalent, statistical convergence and lacunary sequences was studied. Also in [20],

I-asymptotically statistical equivalent and I-asymptotically lacunary statistical equivalent sequences were

examined.

Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation

laws hold, and w(G) and m(G) denote the spaces of all real valued functions and all bounded real functions

f on G respectively. m(G) is a Banach space with the supremum norm ‖f‖∞ = sup{|f(g)| : g ∈ G}.

Nomika [23] showed that, if G is countable amenable group, there exists a sequence {Sn} of finite subsets of

G such that (i) G = ∪∞i=1Sn, (ii) Sn ⊂ Sn+1, n = 1, 2, 3, ..., (iii) limn→∞
|Sng−∩Sn|
|Sn|

= 1, limn→∞
|gSn−∩Sn|
|Sn|

=

1 for all g ∈ G. Here |A| denotes the number of elements in the finite set A. Any sequence of finite subsets

of G satisfying (i), (ii) and (iii) is called a Folner sequence for G.

Amenable semigroups were studied by [1]. The concept of summability in amenable semigroups was

introduced in [13], [14]. In [3], Douglas extended the notion of arithmetic mean to amenable semigroups and

obtained a characterization for almost convergence in amenable semigroups.

In [21], the notions of convergence and statistical convergence, statistical limit point and statistical cluster

point of functions on discrete countable amenable semigroups were introduced.

Nuray F. Rhoades B.E. [22] defined the notions of asymptotically, statistically, almost statistically and

strong almost asymptotically equivalent functions defined on discrete countable amenable semigroups. In

addition to these definitions, they gave some inclusion theorems. Also, they proved that the strong almost

asymptotically equivalence of the functions f(g) and h(g) defined on discrete countable amenable semigroups

does not depend on the particular choice of the Folner sequence.

In [12], the concepts of σ-uniform density of subsets A of the set N of positive integers and corresponding

Iσ-convergence of functions defined on discrete countable amenable semigroups were introduced. Further-

more, for any Folner sequence inclusion relations between Iσ-convergence and invariant convergence also

Iσ-convergence and [Vσ]p-convergence were given. They introduced the concept of Iσ-statistical convergence



Int. J. Anal. Appl. 17 (1) (2019) 16

and Iσ-lacunary statistical convergence of functions defined on discrete countable amenable semigroups. In

addition to these definitions, they gave some inclusion theorems. Also, they made a new approach to the

notions of [V,λ]-summability, σ-convergence and λ-statistical convergence of Folner sequences by using ideals

and introduced new notions, namely, Iσ-[V,λ]-summability, Iσ-λ-statistical convergence of Folner sequences.

Kişi and Güler [11] introduced the concepts of Sσ-asymptotically equivalent, Sσ,λ-asymptotically equiv-

alent, σ-asymptotically lacunary statistical equivalent and strong (σ,θ)-asymptotically equivalent functions

defined on discrete countable amenable semigroups.

The purpose of the study [10] was to extend the notions of I-convergence, I-limit superior and I-limit

inferior, I-cluster point and I-limit point to functions defined on discrete countable amenable semigroups.

Also, a new approach to the notions of [V,λ]-summability and λ-statistical convergence by using ideals and

introduce new notions, namely, I-[V,λ]-summability and I-λ-statistical convergence to functions was defined

on discrete countable amenable semigroups.

This study presents the notion of I-asymptotically lacunary statistical equivalence which is a natural

combination of I-asymptotically equivalence, lacunary statistical equivalence for functions defined on discrete

countable amenable semigroups. We introduce new concepts, and establish certain inclusion theorems.

2. Definitions and Notations

Definition 2.1. [21] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. f ∈ w (G) is said to be convergent to s, for any Folner sequence {Sn} for G,

if for each ε > 0 there exists k0 ∈ N such that |f (g) −s| < ε for all m > k0 and g ∈ G \ Sm.

Definition 2.2. [21] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. f ∈ w(G) is said to be strongly summable to s, for any Folner sequence {Sn}

for G, if

lim
n→∞

1

|Sn|
∑
g∈Sn

|f (g) −s| = 0,

where |Sn| denotes the cardinality of the set Sn.

Definition 2.3. [21] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. f ∈ w(G) is said to be statistically convergent to s, for any Folner sequence

{Sn} for G, if for every ε > 0,

lim
n→∞

1

|Sn|
|{g ∈ Sn : |f (g) −s| ≥ ε}| = 0

The set of all statistically convergent functions will be denoted S(G).



Int. J. Anal. Appl. 17 (1) (2019) 17

Definition 2.4. [22] Let G be a discrete countable amenable semigroup with identity in which both right and

left cancelation laws hold. Two nonnegative functions f,h ∈ w(G) are said to be asymptotically equivalent,

for any Folner sequence {Sn} for G, if for every ε > 0 there exists k0 ∈ N such that∣∣∣∣f (g)h (g) − 1
∣∣∣∣ < ε

for all m > k0 and g ∈ G\Sm. It will be denoted by f ∼ h.

Definition 2.5. [22] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. Two nonnegative functions f,h ∈ w(G) are said to be strong asymptotically

equivalent, for any Folner sequence {Sn} for G, if

lim
n→∞

1

|Sn|
∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ = 0

It will be denoted by f ∼w h.

Definition 2.6. [22] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. Two nonnegative functions f,h ∈ w(G) are said to be statistically equivalent,

for any Folner sequence {Sn} for G, if for every ε > 0,

lim
n→∞

1

|Sn|
|{g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε}| = 0

Definition 2.7. [10] Let G be a discrete countable amenable semigroup with identity in which both right

and left cancelation laws hold. f ∈ w (G) is said to be I-convergent to s for any Folner sequence {Sn} for

G, if for every ε > 0;

{g ∈ Sn : |f (g) −s| ≥ ε}∈I;

i.e., |f (g) −s| < ε a.a.g. The set of all I-convergent sequences will be denoted by I (G).

3. Main Results

Definition 3.1. Let G be a discrete countable amenable semigroup with identity in which both right and left

cancelation laws hold. f ∈ w (G) is said to be I-statistically convergent to s, for any Folner sequence {Sn}

for G, if for each ε > 0 and δ > 0,{
n ∈ N :

1

|Sn|
|{g ∈ Sn : |f (g) −s| ≥ ε}|≥ δ

}
∈I.

The set of all I-statistically convergent Folner sequences will be denoted by SI (G).

Definition 3.2. Let θ be lacunary sequence and G be a discrete countable amenable semigroup with identity

in which both right and left cancelation laws hold. f ∈ w(G) is said to be lacunary statistically convergent to

s, for any Folner sequence {Sn} for G, if for every ε > 0,

lim
r→∞

1

hr
|{g ∈ Sn : |f (g) −s| ≥ ε}| = 0.



Int. J. Anal. Appl. 17 (1) (2019) 18

The set of all lacunary statistically convergent Folner sequences will be denoted Sθ(G).

Definition 3.3. Let θ be lacunary sequence, I ⊆ 2N be an admissible ideal in N and G be a discrete countable

amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (G) is said to be

I-lacunary statistically convergent to s, for any Folner sequence {Sn} for G, if for each ε > 0 and δ > 0,{
r ∈ N :

1

hr
|{g ∈ Sn : |f (g) −s| ≥ ε}|≥ δ

}
∈I.

The set of all I-lacunary statistically convergent sequences will be denoted by SθI (G).

Definition 3.4. Let θ be lacunary sequence, I ⊆ 2N be an admissible ideal in N and G be a discrete countable

amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (G) is said to be

strongly I-lacunary convergent to s or NθI (G)-convergent to s, for any Folner sequence {Sn} for G, if for

each ε > 0, 
r ∈ N : 1hr

∑
g∈Sn

|f (g) −s| ≥ ε


 ∈I.

The set of all strongly I-lacunary convergent sequences will be denoted by NθI (G).

Definition 3.5. Two nonnegative functions f,h ∈ w(G) are said to be asymptotically I-equivalent, for any

Folner sequence {Sn} for G, if every ε > 0,{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}
∈I.

It will be denoted by f ∼I(G) h.

Definition 3.6. Two nonnegative functions f,h ∈ w(G) are said to be I-asymptotically statistical equivalent,

for any Folner sequence {Sn} for G, if for each ε > 0 and δ > 0, the following set{
n ∈ N :

1

|Sn|

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ ≥ δ
}

belongs to I. It will be denoted by f ∼SI(G) h.

For I = Ifin, I-asymptotically statistical equivalence coincides with asymptotically statistical equivalence

for any Folner sequence {Sn} for G.

Definition 3.7. Two nonnegative functions f,h ∈ w(G) are said to be I-asymptotically lacunary statistical

equivalent, for any Folner sequence {Sn} for G, if for every ε > 0 and δ > 0, the following set{
r ∈ N :

1

hr

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ ≥ δ
}

belongs to I. It will be denoted by f ∼S
θ
I(G) h.



Int. J. Anal. Appl. 17 (1) (2019) 19

For I = Ifin, I-asymptotically lacunary statistical equivalence coincides with asymptotically lacunary

statistical equivalence for any Folner sequence {Sn} for G.

Definition 3.8. Two nonnegative functions f,h ∈ w(G) are said to be strongly I-asymptotically lacunary

equivalent, for any Folner sequence {Sn} for G, if for every ε > 0
r ∈ N : 1hr

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε


 ∈I.

It will be denoted by f ∼N
θ
I(G) h.

Definition 3.9. Two nonnegative functions f,h ∈ w(G) are said to be strongly λI (G)-asymptotically equiv-

alent, for any Folner sequence {Sn} for G, if for every ε > 0
n ∈ N : 1λn

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε


 ∈I.

It will be denoted by f ∼VλI (G) h.

Definition 3.10. Two nonnegative functions f,h ∈ w(G) are said to be I-asymptotically λ-statistical equiv-

alent provided that for every ε, δ > 0,{
n ∈ N :

1

λn

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ ≥ δ
}
∈I.

It will be denoted by f ∼SλI (G) h.

Theorem 3.1. Let λ ∈ Λ and I is an admissible ideal in N. If f ∼VλI (G) h then f ∼SλI (G) h.

Proof. Assume that f ∼VλI (G) h and ε > 0. Then,

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ ∑
g∈Sn&|f(g)h(g)−1|≥ε

∣∣∣f(g)h(g) − 1∣∣∣

≥ ε.
∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣

and so,

1

ε.λn

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ 1λn

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ .
Then for any δ > 0, {

n ∈ N : 1
λn

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ}
⊆

{
n ∈ N : 1

λn

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ εδ
}

.

Since right hand belongs to I then left hand also belongs to I and this completes the proof. �



Int. J. Anal. Appl. 17 (1) (2019) 20

Theorem 3.2. Let λ ∈ Λ and I is an admissible ideal in N. If f,h ∈ m (G) are bounded functions and

f ∼SλI (G) h then f ∼VλI (G) h.

Proof. Let f,h ∈ m (G) are bounded functions and f ∼SλI (G) h. Then, there is an M such that∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≤ M

for all k. For each ε > 0,

1

λn

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ = 1λn ∑g∈Sn&|f(g)h(g)−1|≥ε
∣∣∣f(g)h(g) − 1∣∣∣

+
1

λn

∑
g∈Sn&|f(g)h(g)−1|<ε

∣∣∣f(g)h(g) − 1∣∣∣
≤ M

1

λn

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2
}∣∣∣ + ε

2

Then, {
n ∈ N :

1

λn

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε
}

⊆
{
n ∈ N :

1

λn

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2
}∣∣∣ ≥ ε

2M

}
∈I.

Therefore f ∼VλI (G) h. �

Theorem 3.3. If lim inf
λn
|Sn|

> 0 then f ∼SI(G) h implies f ∼SλI (G) h.

Proof. Assume that lim inf
λn
|Sn|

> 0 there exists a δ > 0 such that
λn
|Sn|

≥ δ for sufficiently large n. For given

ε > 0 we have,

1

|Sn|

{
k ≤ |Sn| :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}
⊇

1

|Sn|

{
k ∈ In :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}
,

where In = [n−λn + 1,n]. Therefore,

1
|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ 1|Sn| ∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
≥

λn
|Sn|

.
1

λn

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
≥ δ.

1

λn

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
then for any η > 0 we get{

n ∈ N :
1

λn

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ η
}

⊆
{
n ∈ N : 1|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ ηδ} ∈I,
and this completes the proof. �

Theorem 3.4. If λ = (λn) ∈ ∆ be such that limn→∞
λn
|Sn|

= 1, then f ∼SλI (G) h is subset of f ∼SI(G) h.



Int. J. Anal. Appl. 17 (1) (2019) 21

Proof. Let δ > 0 be given. Since limn→∞
λn
|Sn|

= 1, we can choose m ∈ N such that
∣∣∣∣ λn|Sn| − 1

∣∣∣∣ < δ2 , for all
n ≥ m. Now observe that, for ε > 0

1

|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ = 1|Sn|
∣∣∣{k ≤ |Sn|−λn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣

+
1

|Sn|

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
≤
|Sn|−λn
|Sn|

+
1

|Sn|

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
≤ 1 −

(
1 −

δ

2

)
+

1

|Sn|

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣
=

δ

2
+

1

|Sn|

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ,
for all n ≥ m. Hence,{

g ∈ Sn :
1

|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ
}

⊂
{
g ∈ Sn :

1

|Sn|

∣∣∣{k ∈ In : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ2
}
∪{1, 2, ...,m}

If f ∼SλI (G) h, then the set on the right hand side belongs to I and so the set on the left hand side also

belongs to I. This shows that f ∼SI(G) h. �

Definition 3.11. Two nonnegative functions f,h ∈ w(G) are said to be strongly Cesáro I-asymptotically

equivalent, provided that for every ε > 0,
g ∈ Sn : 1|Sn|

∑
1≤k≤|Sn|

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε


 ∈I.

It will be denoted by f ∼[C1(I)](G) h.

Theorem 3.5. If f ∼VλI (G) h, then f ∼[C1(I)](G) h.

Proof. Assume that f ∼VλI (G) h and ε > 0. Then,

1
|Sn|

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ = 1|Sn| |Sn|−λn∑
k=1

∣∣∣f(g)h(g) − 1∣∣∣ + 1|Sn| ∑
k∈In,g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣

≤
1

λn

|Sn|−λn∑
k=1

∣∣∣f(g)h(g) − 1∣∣∣ + 1λn ∑k∈In,g∈Sn
∣∣∣f(g)h(g) − 1∣∣∣

≤
2

λn

∑
k∈In,g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣



Int. J. Anal. Appl. 17 (1) (2019) 22

and so, 
g ∈ Sn : 1|Sn|

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε


 ⊆


g ∈ Sn : 1λn

∑
k∈In,g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε2




belongs to I. Hence f ∼[C1(I)](G) h. �

Theorem 3.6. Let λ ∈ Λ and I is an admissible ideal in N. If f,h ∈ m (G) are bounded and f ∼SλI (G) h

then f ∼C1(I)(G) h.

Proof. Suppose that f,h ∈ m (G) and f ∼SλI (G) h. Since f,h ∈ m (G), they are bounded, assume that∣∣∣f(g)h(g) − 1∣∣∣ < M for all g. Let ε > 0 be given and set
Ln =

{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε2

}
.

For each ε > 0,
1

λn

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ = 1λn ∑g∈Sn
∣∣∣f(g)h(g) − 1∣∣∣ + 1λn ∑g∈Sn\Ln

∣∣∣f(g)h(g) − 1∣∣∣
≤ M

1

λn

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + ε2 .
Then, {

n ∈ N :
1

λn

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε
}

⊆
{
n ∈ N :

1

λn

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2
}∣∣∣ ≥ ε

2M

}
∈I.

Therefore, f ∼VλI (G) h. �

Theorem 3.7. If f ∼C1(I)(G) h, then, f ∼S(G) h.

Proof. Let f ∼C1(I)(G) h, and ε > 0 given. Then,

1
|Sn|

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ 1|Sn| ∑
g∈Sn&|f(g)h(g)−1|≥ε

∣∣∣f(g)h(g) − 1∣∣∣

≥ ε.
∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣

and so
1

ε. |Sn|
∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ 1|Sn|

∣∣∣∣
{
k ≤ |Sn| :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ .
So for a given δ > 0, {

n ∈ N : 1|Sn|
∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ}

⊆

{
n ∈ N : 1|Sn|

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ εδ
}
∈I.

Therefore f ∼S(G) h. �



Int. J. Anal. Appl. 17 (1) (2019) 23

Theorem 3.8. Let f, h ∈ m (G). If f ∼S(G) h then f ∼C1(I)(G) h.

Proof. Suppose that f, h ∈ m (G) and f ∼Sλ(G) h. Then, there is a M such that
∣∣∣f(g)h(g) − 1∣∣∣ ≤ M for all k.

Given ε > 0, we have

1
|Sn|

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ = 1|Sn| ∑
g∈Sn&|f(g)h(g)−1|≥ε

∣∣∣f(g)h(g) − 1∣∣∣

+ 1|Sn|
∑

g∈Sn&|f(g)h(g)−1|<ε

∣∣∣f(g)h(g) − 1∣∣∣

≤ 1|Sn|.M
∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣

+ 1|Sn|.ε
∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ < ε}∣∣∣

≤
M

|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + ε.
Then, for any δ > 0,{

n ∈ N : 1|Sn|
∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ δ
}

⊆
{
n ∈ N : 1|Sn|

∣∣∣{k ≤ |Sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δM
}
∈I.

Therefore f ∼C1(I)(G) h. �

Theorem 3.9. Let f,h ∈ w(G) two nonnegative functions. Then,

(a) If f ∼N
θ
I(G) h, then f ∼S

θ
I(G) h,

(b) If f,h ∈ m (G) and f ∼S
θ
I(G) h, then f ∼N

θ
I(G) h.

Proof. (a). Let ε > 0 and f ∼N
θ
I(G) h. Then, we can write∑

g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε.

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣
and so

1

εhr

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ 1hr

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε

}∣∣∣∣ .
Then, for any δ > 0 {

r ∈ N : 1
hr

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ}

⊆

{
r ∈ N : 1

hr

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε.δ
}
∈I.



Int. J. Anal. Appl. 17 (1) (2019) 24

Hence, we have f ∼S
θ
I(G) h.

(b) Suppose that f,h ∈ m (G) and f ∼S
θ
I(G) h. Since f,h ∈ m (G), they are bounded, assume that∣∣∣f(g)h(g) − 1∣∣∣ < M for all g.

1

hr

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ = 1hr ∑g∈Sn
∣∣∣f(g)h(g) − 1∣∣∣ + 1hr ∑g∈Sn\Ln

∣∣∣f(g)h(g) − 1∣∣∣
≤

M

hr

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2
}∣∣∣ + ε

2
.

And define the sets

D1 =


r ∈ N : 1hr

∑
g∈Sn

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε




and

D2 =

{
r ∈ N :

1

hr

∣∣∣∣
{
g ∈ Sn :

∣∣∣∣f (g)h (g) − 1
∣∣∣∣ ≥ ε2

}∣∣∣∣ ≥ ε2M
}

.

If r /∈ D2, then 1hr
∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2

}∣∣∣ < ε
2M

.

Also we can get

1
hr

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≤ Mhr
∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2

}∣∣∣ + ε
2

<
ε

2
+
ε

2
= ε.

Thus, r /∈ D1. Consequently, we have{
r ∈ N : 1

hr

∑
g∈Sn

∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε
}

⊆
{
r ∈ N : 1

hr

∣∣∣{g ∈ Sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2
}∣∣∣ ≥ ε

2M

}
∈I.

Therefore, f ∼N
θ
I(G) h. This completes the proof. �

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	1. Introduction
	2. Definitions and Notations
	3. Main Results
	References