# On Inclusion Relations between Some Sequence Spaces

Authors: R. Çolak; Ç. A. Bektaş; H. Altınok; S. Ercan
Journal: International Journal of Analysis
 (2016)

---

## Abstract

We determine the relations between the classesS
^
λ of almost λ-statistically convergent sequences and the relations between the classes V
^
,
λ of strongly almost V
,
λ-summable sequences for various sequences λ, μ in the class Λ. Furthermore we also give the relations between the classes S
^
λ of almost λ-statistically convergent sequences and the classes V
^
,
λ of strongly almost V
,
λ-summable sequences for various sequences λ
,
μ
∈
Λ.


---

## Body

## 1. Introduction

A sequencex
=
x
k of real (or complex) numbers is said to bestatistically convergentto the number L if for every ε
>
0
(1)
lim
n
→
∞
⁡
1
n
k
≤
n
:
x
k
-
L
≥
ε
=
0
.In this case, we write S
-
lim
⁡
x
=
L or x
k
→
L
S and S denotes the set of all statistically convergent sequences.A sequencex
=
x
k of real (or complex) numbers is said to bealmost statistically convergentto the number L if for every ε
>
0
(2)
lim
n
→
∞
⁡
1
n
k
≤
n
:
x
k
+
m
-
L
≥
ε
=
0
uniformly in
m
.In this case, we write S
^
-
lim
⁡
x
=
L or x
k
→
L
S
^ and S
^ denotes the set of all almost statistically convergent sequences [1].Letλ
=
λ
n be a nondecreasing sequence of positive real numbers tending to ∞ such that (3)
λ
n
+
1
≤
λ
n
+
1
,
λ
1
=
1
.The set of all such sequences will be denoted by Λ
.The generalized de la Vallée-Poussin mean is defined by(4)
t
n
x
=
1
λ
n
∑
k
∈
I
n
x
k
, where I
n
=
n
-
λ
n
+
1
,
n.A sequencex
=
x
k is said to be V
,
λ-summable to a number L (see [2]) if (5)
t
n
x
⟶
L
as
n
⟶
∞
. If λ
n
=
n for each n
∈
N, then V
,
λ-summability reduces to C
,
1-summability.We write(6)
C
,
1
=
x
=
x
k
:
lim
n
→
∞
⁡
1
n
∑
k
=
1
n
x
k
-
L
=
0
for some
L
,
V
,
λ
=
x
=
x
k
:
lim
n
→
∞
⁡
1
λ
n
∑
k
∈
I
n
x
k
-
L
=
0
for some
Lfor the sets of sequences x
=
x
k which are strongly Cesàro summable and strongly V
,
λ-summable to L; that is, x
k
→
L
C
,
1 and x
k
→
L
V
,
λ, respectively.Savaš [1] defined the following sequence space: (7)
V
^
,
λ
=
x
=
x
k
:
l
i
m
n
→
∞
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
L
=
0
for some
L
,
uniformly  in
mfor the sets of sequences x
=
x
k which are strongly almost V
,
λ-summable to L; that is, x
k
→
L
V
,
λ. We will write V
^
,
λ
∞
=
V
^
,
λ
∩
l
∞
.Theλ-statistical convergence was introduced by Mursaleen in [3] as follows.Letλ
=
λ
n
∈
Λ. A sequence x
=
x
k is said to be λ-statistically convergent or S
λ-convergent to L if for every ε
>
0
(8)
l
i
m
n
→
∞
1
λ
n
k
∈
I
n
:
x
k
-
L
≥
ε
=
0
,where I
n
=
n
-
λ
n
+
1
,
n. In this case we write S
λ
-
lim
⁡
x
=
L or x
k
→
L
(
S
λ
), and S
λ
=
x
=
x
k
:
S
λ
-
lim
⁡
x
=
L
for some
L.The sequencex
=
x
k is said to be λ-almost statistically convergent if there is a complex number L such that (9)
lim
n
→
∞
⁡
1
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
ε
=
0
,
uniformly in
m
.In this case, we write S
^
λ
-
lim
⁡
x
=
L or x
k
→
L
(
S
^
λ
) and S
^
λ denotes the set of all λ-almost statistically convergent sequences. If we choose λ
n
=
n for all n, then λ-almost statistical convergence reduces to almost statistical convergence [1].


## 2. Main Results

Throughout the paper, unless stated otherwise, by “for alln
∈
N
n
o” we mean “for all n
∈
N except finite numbers of positive integers” where N
n
o
=
n
o
,
n
o
+
1
,
n
o
+
2
,
… for some n
o
∈
N
=
1,2
,
3
,
….Theorem 1.
Letλ
=
λ
n and μ
=
μ
n be two sequences in Λ such that λ
n
≤
μ
n for all n
∈
N
n
o
. Consider the following:(i)
If(10)
liminf
n
→
∞
⁡
λ
n
μ
n
>
0
thenS
^
μ
⊆
S
^
λ.
(ii)
If(11)
lim
n
→
∞
⁡
λ
n
μ
n
=
1
thenS
^
λ
⊆
S
^
μ.Proof.
(i) Suppose thatλ
n
≤
μ
n for all n
∈
N
n
o and let (10) be satisfied. Then I
n
⊂
J
n so that for ε
>
0 we may write(12)
k
∈
J
n
:
x
k
+
m
-
L
≥
ε
≥
k
∈
I
n
:
x
k
+
m
-
L
≥
εand therefore we have (13)
1
μ
n
k
∈
J
n
:
x
k
+
m
-
L
≥
ε
≥
λ
n
μ
n
1
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
εfor all n
∈
N
n
o, where J
n
=
n
-
μ
n
+
1
,
n
. Now taking the limit as n
→
∞ uniformly in m in the last inequality and using (10) we get x
k
→
L
S
^
μ
⇒
x
k
→
L
S
^
λ so that S
^
μ
⊆
S
^
λ.
(ii) Letx
k
∈
S
^
λ and (11) be satisfied. Since I
n
⊂
J
n, for ε
>
0, we may write (14)
1
μ
n
k
∈
J
n
:
x
k
+
m
-
L
≥
ε
=
1
μ
n
n
-
μ
n
+
1
≤
k
≤
n
-
λ
n
:
x
k
+
m
-
L
≥
ε
+
1
μ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
ε
≤
μ
n
-
λ
n
μ
n
+
1
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
ε
≤
1
-
λ
n
μ
n
+
1
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
εfor all n
∈
N
n
o. Since lim
n
⁡
λ
n
/
μ
n
=
1 by (11) and since x
=
x
k
∈
S
^
λ the first term and second term of right hand side of above inequality tend to 0 as n
→
∞ uniformly in m. This implies that 1
/
μ
n
k
∈
J
n
:
x
k
+
m
-
L
≥
ε
→
0 as n
→
∞ uniformly in m
. Therefore S
^
λ
⊆
S
^
μ.From Theorem1 we have the following result.Corollary 2.
Letλ
=
λ
n and μ
=
μ
n be two sequences in Λ such that λ
n
≤
μ
n for all n
∈
N
n
o. If (11) holds then S
^
λ
=
S
^
μ.If we takeμ
=
μ
n
=
n in Corollary 2 we have the following result.Corollary 3.
Letλ
=
λ
n
∈
Λ
. If lim
n
⁡
λ
n
/
n
=
1 then we have S
^
λ
=
S
^
.Theorem 4.
Letλ
=
λ
n and μ
=
μ
n
∈
Λ and suppose that λ
n
≤
μ
n for all n
∈
N
n
o. Consider the following:(i)
If (10) holds then V
^
,
μ
⊆
V
^
,
λ.
(ii)
If (11) holds then V
^
,
λ
∞
⊆
V
^
,
μ.Proof.
(i) Suppose thatλ
n
≤
μ
n for all n
∈
N
n
o. Then I
n
⊆
J
n so that we may write (15)
1
μ
n
∑
k
∈
J
n
x
k
+
m
-
L
≥
1
μ
n
∑
k
∈
I
n
x
k
+
m
-
L for all n
∈
N
n
o. This gives that (16)
1
μ
n
∑
k
∈
J
n
x
k
+
m
-
L
≥
λ
n
μ
n
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
L
.Then taking limit as n
→
∞, uniformly in m in the last inequality, and using (10) we obtain x
k
→
L
V
^
,
μ
⇒
x
k
→
L
V
^
,
λ.
(ii) Letx
=
x
k
∈
V
^
,
λ
∞ be any sequence. Suppose that x
k
→
L
V
^
,
μ and that (11) holds. Since x
=
x
k
∈
l
∞ then there exists some M
>
0 such that x
k
+
m
-
L
≤
M for all k and m. Since λ
n
≤
μ
n so that 1
/
μ
n
≤
1
/
λ
n, and I
n
⊂
J
n for all n
∈
N
n
o, we may write (17)
1
μ
n
∑
k
∈
J
n
x
k
+
m
-
L
=
1
μ
n
∑
k
∈
J
n
-
I
n
x
k
+
m
-
L
+
1
μ
n
∑
k
∈
I
n
x
k
+
m
-
L
≤
μ
n
-
λ
n
μ
n
M
+
1
μ
n
∑
k
∈
I
n
x
k
+
m
-
L
≤
1
-
λ
n
μ
n
M
+
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
Lfor every n
∈
N
n
o. Since lim
n
⁡
λ
n
/
μ
n
=
1 by (11) and since x
k
→
L
V
^
,
λ the first term and the second term of right hand side of above inequality tend to 0 as n
→
∞, uniformly in m (note that 1
-
λ
n
/
μ
n
≥
0 for all n
∈
N
n
o). Then we get x
k
→
L
V
^
,
λ
⇒
x
k
→
L
V
^
,
μ. Since x
=
x
k
∈
V
^
,
λ
∞ is an arbitrary sequence we obtain V
^
,
λ
∞
⊆
V
^
,
μ.Since clearly (11) implies (10) from Theorem 4 we have the following result.Corollary 5.
Letλ
,
μ
∈
Λ such that λ
n
≤
μ
n for all n
∈
N
n
o. If (11) holds then V
^
,
λ
∞
=
V
^
,
μ
∞.Theorem 6.
Letλ
,
μ
∈
Λ such that λ
n
≤
μ
n for all n
∈
N
n
o. Consider the following:(i)
If (10) holds then (18)
x
k
⟶
L
V
^
,
μ
⟹
x
k
⟶
L
S
^
λ
and the inclusionV
^
,
μ
⊂
S
^
λ holds for some λ
,
μ
∈
Λ.
(ii)
Ifx
k
∈
l
∞ and x
k
→
L
S
^
λ then x
k
→
L
V
^
,
μ, whenever (11) holds.
(iii)
If (11) holds then l
∞
∩
S
^
λ
=
V
^
,
μ
∞.Proof.
(i) Letε
>
0 be given and let x
k
→
L
V
^
,
μ. Now for every ε
>
0 we may write (19)
∑
k
∈
J
n
x
k
+
m
-
L
≥
∑
k
∈
I
n
x
k
+
m
-
L
≥
∑
k
∈
I
n
x
k
+
m
-
L
≥
ε
x
k
+
m
-
L
≥
ε
k
∈
I
n
:
x
k
+
m
-
L
≥
ε so that (20)
1
μ
n
∑
k
∈
J
n
x
k
+
m
-
L
≥
λ
n
μ
n
1
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
ε
εfor all n
∈
N
n
o. Then taking limit as n
→
∞, uniformly in m in the last inequality, and using (10) we obtain that x
k
→
L
S
^
λ whenever x
k
→
L
V
^
,
μ. Since x
=
x
k
∈
V
^
,
μ is an arbitrary sequence we obtain that V
^
,
μ
⊂
S
^
λ.
(ii) Suppose thatx
k
→
L
S
^
λ and x
=
x
k
∈
l
∞. Then there exists some M
>
0 such that x
k
+
m
-
L
≤
M for all k and m
. Since 1
/
μ
n
≤
1
/
λ
n, then for every ε
>
0 we may write(21)
1
μ
n
∑
k
∈
J
n
x
k
+
m
-
L
=
1
μ
n
∑
k
∈
J
n
-
I
n
x
k
+
m
-
L
+
1
μ
n
∑
k
∈
I
n
x
k
+
m
-
L
≤
μ
n
-
λ
n
μ
n
M
+
1
μ
n
∑
k
∈
I
n
x
k
+
m
-
L
≤
1
-
λ
n
μ
n
M
+
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
L
≤
1
-
λ
n
μ
n
M
+
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
L
≥
ε
x
k
+
m
-
L
+
1
λ
n
∑
k
∈
I
n
x
k
+
m
-
L
<
ε
x
k
+
m
-
L
≤
1
-
λ
n
μ
n
M
+
M
λ
n
k
∈
I
n
:
x
k
+
m
-
L
≥
ε
+
εfor all n
∈
N
n
o
. Using (11) we obtain that x
k
→
L
V
^
,
μ whenever x
k
→
L
S
^
λ
. Since x
=
x
k
∈
l
∞
∩
S
^
λ is an arbitrary sequence we obtain l
∞
∩
S
^
λ
⊆
V
^
,
μ.
(iii) The proof follows from (i) and (ii), so we omit it.From Theorem1(i) and Theorem 6(i) we obtain the following result.Corollary 7.
Ifliminf
n
→
∞
⁡
λ
n
/
μ
n
>
0 then S
^
μ
∩
V
^
,
μ
⊂
S
^
λ.If we takeμ
n
=
n for all n in Theorem 6 then we have the following results, because lim
n
→
∞
⁡
λ
n
/
μ
n
=
1 implies that liminf
n
→
∞
⁡
λ
n
/
μ
n
=
1
>
0; that is, (11) ⇒ (10).Corollary 8.
Iflim
n
→
∞
⁡
λ
n
/
n
=
1 then(i)
ifx
k
∈
l
∞ and x
k
→
L
S
^
λ then x
k
→
L
C
,
1,
(ii)
ifx
k
→
L
C
,
1 then x
k
→
L
S
^
λ.



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*Source: 7283527-2016-08-03.xml*