R E S E A R C H
Open Access
f
(
λ
,
μ
)
-statistical convergence of order
α
for
double sequences
Mahmut I¸sik
1*and Yavuz Altin
2*Correspondence:
1Faculty of Education, Harran
University, ¸Sanlıurfa, Turkey Full list of author information is available at the end of the article
Abstract
New concepts offλ,μ-statistical convergence for double sequences of order
α
and strongfλ,μ-Cesàro summability for double sequences of orderα
are introduced for sequences of (complex or real) numbers. Furthermore, we give the relationship between the spacesw2˜
α,0(f,
λ
,μ
),w 2˜
α(f,
λ
,μ
) andwα2˜,∞(f,λ
,μ
). Then we express the properties of strongfλ,μ-Cesàro summability of orderβ
which is related to strongfλ,μ-Cesàro summability of order
α
. Also, some relations betweenfλ,μ-statistical convergence of orderα
and strongfλ,μ-Cesàro summability of orderα
are given.MSC: 40A05; 40C05; 46A45
Keywords: double sequences; statistical convergence; Cesàro summability
1 Introduction
The first idea of statistical convergence goes back to the first edition of the famous Zyg-mund’s monograph []. The statistical convergence was introduced for real and complex sequences by Steinhaus []. Fast [] extended the usual concept of sequential limit and called it statistical convergence. Schoenberg [] called it as D-Convergence. The idea de-pends on a certain density of subsets ofN. The natural density (or asymptotic density) of a setA⊂ Nis defined byδ(A) =limn→∞n|{k≤n:k∈A}|if the limit exists, where
|A(n)|is cardinality of the setA(n) (see []). A sequencex= (xk) of complex numbers is said to be statistically convergent to some numberifδ({k∈N:|xk–| ≥ε}) has natural density zero forε> .is necessarily unique, which is statistical limit of (xk), and written asS-limxk=. The space of all statistically convergent sequences is denoted byS (see [–]).
The order of statistical convergence of a sequence of positive linear operators was given by Gadjiev and Orhan [], and after that Çolak [] introduced statistical convergence of orderαand strongp-Cesàro summability of orderα.
Statistical convergence was introduced for double sequences by Mursaleen and Edely []. Besides this topic was studied by many authors (such as [, , ]). For some further works in this direction, we refer to [–].
The concepts of convergence and statistical convergence for double sequence can be expressed as follows.
Letsdenote the space of all double sequences, and let
∞,candcbe the linear spaces of bounded, convergent and null sequences x= (xjk) with complex terms, respectively, normed byx(∞,)=supj,k|xjk|, wherej,k∈N={, , . . .}.
A double sequencex= (xj,k)∞j,k=has Pringsheim limitprovided that for everyε> there existsN∈Nsuch that|xj,k–|<εwheneverj,k>N. In this case, we writeP-limx=
[].
x= (xj,k)∞j,k=is bounded if there exists a positive numberMsuch that|xj,k|<Mfor allj andk, that is,x=supj,k≥|xj,k|<∞.
LetK⊆N×NandK(m,n) ={(j,k) :j≤m,k≤n}. The double natural density ofKis defined by
δ(K) =P-lim m,n
mnK(m,n) if the limit exists.
A double sequencex= (xjk)j,k∈Nis said to be statistically convergent toif for everyε> the set{(j,k) :j≤m,k≤n:|xjk–| ≥ε}has double natural density zero []. In this case, one can writest-limx=, and we denote the collection of all statistically convergent double sequences by st. Recently, Çolak and Altin [] introduced double statistically convergent of orderα, and they examined some inclusion relations.
The idea of a modulus function was introduced in by Nakano []. Later, Ruckle [] and Maddox [] used this concept to construct some sequence spaces. Let us remind modulus function.
f : [,∞)→[,∞) is called a modulus function if
. f(x) = if and only ifx= ,
. f(x+y)≤f(x) +f(y)for everyx,y∈R+,
. f is increasing,
. f is continuous from the right at.
Hence, f must be continuous everywhere on [,∞). A modulus function may be bounded or unbounded. For example, f(x) = +xx is bounded, but f(x) =xp, <p≤ is unbounded.
Aizpuruet al.[] introduced and discussed the concepts off-statistical convergence andf-statistically Cauchy sequences, a single sequence of numbers, wheref is an un-bounded modulus function. Bhardwaj and Dhawan [] continued this work and defined f-statistical convergence of orderα. This new idea was introduced by Borgohain and Savaş [] under the name of ’fλ-statistical convergence’. Aizpuruet al.also studied these
con-cepts for double sequences []. Mursaleen [] introducedλ-statistical convergence as an extension of (V,λ)-summability of Leindler [] with the help of a non-decreasing se-quence,λ= (λn) being a non-decreasing sequence of positive numbers tending to∞with λn+≤λn+ ,λ= . The generalized de la Vallee-Poussin mean is defined by
tn(x) =
λn
k∈In
xk,
whereIn= [n–λn+ ,n].
2 fλ,μ-double statistical convergence of order
α
In this section, we introducefλ,μ-double statistical convergence of orderαfor double
se-quences.
Throughout this paper, we takes,t,u,v∈(, ] as otherwise indicated. We will writeα
instead of (s,t) andβinstead of (u,v). Also, we define the following:
αβ ⇐⇒ s≤u and t≤v, α≺β ⇐⇒ s<u and t<v, α∼=β ⇐⇒ s=u and t=v, α∈(, ] ⇐⇒ s,t∈(, ],
β∈(, ] ⇐⇒ u,v∈(, ],
α∼= in cases=t= ,
β∼= in caseu=v= ,
α in cases> and t> .
Furthermore, we write Sα˜(f,λ,μ) to denote S(s,t)(f,λ,μ) and Sβ(f,λ,μ) to denote S(u,v)(f,λ,μ) in the section below.
We begin with the following definitions.
Letλ= (λn) andμ= (μm) be two non-decreasing sequences of positive real numbers tending to∞withλn+≤λn+ ,λ= ;μn+≤μn+ ,μ= andα∈(, ] be given.
LetK⊆N×Nbea two-dimensional set of positive integers andf be an unbounded modulus function. Thenδfα˜(λ,μ)-doubledensityofKis defined as
δfα˜(K) = lim n,m→∞
f(λs
nμtm)
f(j,k)∈In×Im: (i,j)∈K if the limit exists.
Definition . Letλ= (λn) andμ= (μm) be two non-decreasing sequences of positive
real numbers as above andα∈(, ] be given.
(xjk) is said to befλ,μ-statistically convergent of orderαif there is a complex number
such that, for everyε> ,
lim n,m→∞
f(λs
nμtm)
f(j,k)∈In×Im:|xjk–| ≥ε= .
In this case we writeS
˜
α(f,λ,μ)-limj,kxjk=, and we denote the set of allfλ,μ-statistically
convergent double sequences of orderαbyS
˜
α(f,λ,μ), wheref is an unbounded modulus
function.
In the case of f(x) =x,α ∼= andλn=n,μm =m, fλ,μ-statistical convergence of
or-derαreduces to the statistical convergence of double sequences []. Ifx= (xjk) isfλ,μ
-statistically convergent of orderαto the number, thenis determined uniquely.fλ,μ
defined forα. For this, let us definex= (xjk) as follows:
xjk= ⎧ ⎨ ⎩
, ifj+keven,
, ifj+kodd.
Sincelimt→∞f(t)t > , we have
lim n,m→∞
f(λs
nμtm)
f(j,k)∈In×Im:|xjk– | ≥ε≤ lim n,m→∞
f([|λs
nμtm|]) + f(λs
nμtm) =
and
lim n,m→∞
f(λs
nμtm)
(j,k)∈In×Im:|xjk– | ≥ε≤ lim n,m→∞
f([|λsnμtm|]) + f(λs
nμtm) =
forα, that is,s> andt> , so thatx= (xjk) isfλ,μ-statistically convergent of orderα
both to and , i.e.,S
˜
α(f,λ,μ)-limxjk= andSα˜(f,λ,μ)-limxjk= . But this is impossible. Theorem . Let f be an unbounded modulus function andα∈(, ].Let x= (xjk),y= (yjk) be any two sequences of complex numbers.Then
(i) IfSα˜(f,λ,μ)-limxjk=andc∈C,thenSα˜(f,λ,μ)-limcxjk=c; (ii) IfS
˜
α(f,λ,μ)-limxjk=oandSα˜(f,λ,μ)-limyjk=,then
S
˜
α(f,λ,μ)-lim(xjk+yjk) =+.
Theorem . Let f be an unbounded modulus function andα˜,β˜be two real numbers such thatαβ.Then S
˜
α(f,λ,μ)⊆Sβ˜(f,λ,μ)and strict inclusion may occur.
Proof Letα˜,β˜∈(, ] be given such thatα˜≤ ˜β. Sincef is increasing, we have
f(λu
nμvm)
f(j,k)∈In×Im:|xjk–| ≥ε
≤
f(λs nμtm)
f(j,k)∈In×Im:|xjk–| ≥ε
for everyε> , and this givesS
˜
α(f,λ,μ)⊆Sβ˜(f,λ,μ). To show that the strict inclusion may
occur, consider a sequencex= (xjk) defined by
xjk= ⎧ ⎨ ⎩
jk, ifn– [|λn|] + ≤j≤nandm– [|μm|] + ≤k≤m,
, otherwise
and we takef(x) =xp, ( <p≤) and hencex∈Sβ˜(f,λ,μ) forβ˜∈(, ], (i.e.,<u≤ and
<v≤ ), butx∈/S
˜
α(f,λ,μ) forα˜ ∈(,
] (i.e., <s≤
and <t≤
).
The following results can be easily derived from Theorem ..
Corollary . If x= (xjk)is fλ,μ-statistically convergent of orderαto,for someαsuch that
Corollary . Letα,β∈(, ]be given.Then
(i) Sα(f,λ,μ) =Sβ(f,λ,μ)ifα∼=β. (ii) Sα(f,λ,μ) =S(f,λ,μ)ifα∼= .
3 Strongly double Cesàro summability of order
α
defined by a modulus functionIn this section, we give the relationships between the spaceswα˜,(f,λ,μ),wα˜(f,λ,μ) and wα˜,∞(f,λ,μ).
Definition . Letf be a modulus function andα˜ be a positive real number. We have
wα˜,(f,λ,μ) =
x= (xjk)∈s: lim n,m→∞
(λnμm)α˜
j∈Jn
k∈In
f|xjk|=
,
wα˜(f,λ,μ) =
x= (xjk)∈s: lim n,m→∞
(λnμm)α˜
j∈Jn
k∈In
f|xjk–|=
,
wα˜,∞(f,λ,μ) =
x= (xjk)∈s:sup n,m
(λnμm)α˜
j∈Jn
k∈In
f|xjk|
<∞
.
Theorem .
(i) Letf be a modulus function.Forα˜,we havewα˜,(f,λ,μ)⊂wα˜,∞(f,λ,μ). (ii) Letf be a modulus function.Forα˜ ,we havewα˜(f,λ,μ)⊂wα˜,∞(f,λ,μ).
Proof (i) The proof of (i) is trivial.
(ii) Letx∈wα˜(f,λ,μ). By the definition of modulus function (ii) and (iii), we have
(λnμm)α˜
j∈Jn
k∈In
f|xjk|≤ (λnμm)α˜
j∈Jn
k∈In
f|xjk–|+f||
(λnμm)α˜
j∈Jn
k∈In
,
and sinceα˜ andx∈w
˜
α(f,λ,μ), we havex∈wα˜,∞(f,λ,μ), which completes the proof. Theorem . For any modulus function f andα˜ ,we have w
˜
α(λ,μ)⊂ wα˜(f,λ,μ),
w
˜
α,(λ,μ)⊂wα˜,(f,λ,μ)and wα˜,∞(λ,μ)⊂wα˜,∞(f,λ,μ). Proof We give the proof only whenw
˜
α,∞(λ,μ)⊂wα˜,∞(f,λ,μ) and the rest of cases will follow similarly. Letx∈w
˜
α,∞(λ,μ), so that
sup n,m
(λnμm)α˜
j∈Jn
k∈In
|xjk|<∞.
Letε> and chooseδwith <δ< such thatf(t) <εfor ≤t<δ. Now we write
(λnμm)α˜
j∈Jn
k∈In
f|xjk|=
+
where the first summation is over|xjk| ≤δ and the second is over|xjk|>δ. Then≤
ε·
(λnμm)α˜– and, for|xjk|>δ, we use the fact that |xjk|<
|xjk| δ < +
|xjkδ|,
where [|t|] denotes the integer part oft. Givenε> , by the definition off, we have
f|xjk|≤
+|xjk|
δ
f()≤f()|xjk|
δ
for|xjk|>δand hence≤f()δ– j∈Jn
k∈In|xjk|, which together with
≤ε(λnμm)α˜–
yields
(λnμm)α˜
j∈Jn
k∈In
f|xjk|≤ε·
(λnμm)α˜–
+ f()δ–
(λnμm)α˜
j∈Jn
k∈In |xjk|.
Sinceα˜≥ andx∈wα˜,∞(λ,μ), we havex∈wα˜,∞(f,λ,μ) and the proof is complete. Theorem . Let f be a modulus function f andα˜ .Iflimt→∞f(t)t > ,then wα˜(f,
λ,μ)⊂w
˜
α(λ,μ).
Proof Following the proof of Proposition of Maddox [], we havel=limt→∞f(t)t = inf{f(t)t :t> }. By the definition ofl, we havef(t)≥ltfor allt≥. Sincel> , we get t≤l–f(t) for allt≥, and so
(λnμm)α˜
j∈Jn
k∈In
|xjk–| ≤l– (λnμm)α˜
j∈Jn
k∈In
f|xjk–|
,
from where it follows thatx∈w
˜
α(f,λ,μ) wheneverx∈wα˜(λ,μ).
Theorem . For any modulus f such thatlimt→∞f(t)t > andα˜ .Then wα˜(λ,μ) =
wα˜(f,λ,μ).
4 Relation betweenfλ,μ-statistical convergence of order
α
and strongly doubleCesàro summability of order
α
defined by a modulus functionIn this section, we give the relationship between the strongfλ,μ-Cesàro summability of
orderαandfλ,μ-statistical convergence of orderβ.
Lemma . Let f be an unbounded function such that there is a positive constant c such that f(xy)≥cf(x)f(y)for all x≥,y≥ [].
Theorem . Let≺ ˜α ˜βand f be an unbounded modulus function such that there
is a positive constant c such that f(xy)≥cf(x)f(y)for all x≥,y≥andlimt→∞f(t)t > .If a sequence x= (xjk)is strongly fλ,μ-Cesàro summable of orderα˜ with respect to f to,then
Proof For any sequencex= (xjk) andε> , using the definition of modulus function (ii) and (iii), we have
j∈Jn
k∈In
f|xjk–L|≥f j∈Jn
k∈In |xjk–|
≥f(j,k)∈In×Im:|xjk–| ≥εε
≥cf(j,k)∈In×Im:|xjk–| ≥εf(ε)
and sinceα˜ ˜β
nsmt
n
j= m
k=
f|xjk–|
≥
nsmtcf(j,k)∈In×Im:|xjk–| ≥εf(ε)
≥
numvcf(j,k)∈In×Im:|xjk–| ≥εf(ε)
=
numvf(numv)cf(j,k)∈In×Im:|xjk–| ≥εf(ε)
fnumv,
where, using the fact thatlimt→∞f(t)t > andx∈wα˜(f,λ,μ), it follows thatx∈S
˜
β(λ,μ)
and the proof is complete.
If we takeβ∼=αin Theorem ., we have the following.
Corollary . Let f be an unbounded modulus function f(xy)≥cf(x)f(y),where c is a positive constant for all x≥,y≥andlimt→∞f(t)
t > andα˜ ∈(, ].If a sequence is strongly fλ,μ-Cesàro summable of orderα˜ with respect to f to,then it is fλ,μ-statistically
convergent of orderα˜to.
5 Conclusions
In this study, we define fλ,μ-statistical convergence for double sequences of orderα,
wheref is an unbounded modulus function. Besides this we also study strongfλ,μ-Cesàro
summability for double sequences of orderα and give inclusion relations. These results are the generalizations of the studies by Meenakshiet al.[].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Author details
1Faculty of Education, Harran University, ¸Sanlıurfa, Turkey.2Department of Mathematics, Fırat University, Elazı ˘g, Turkey.
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