On λ statistical convergence of order α of sequences of function

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R E S E A R C H

Open Access

On

λ

-statistical convergence of order

α

of

sequences of function

Mikail Et

1

_{, Muhammed Çınar}

2*

_{and Murat Karaka¸s}

3

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,} Mu¸sAlparslan University, Mu¸s, Turkey

Full list of author information is available at the end of the article

Abstract

In this study we introduce the concept ofSα_λ(f)-statistical convergence of sequences of real valued functions. Also some relations betweenSα_λ(f)-statistical convergence and strongwβ_λ_p(f)-summability are given.

MSC: 40A05; 40C05; 46A45

Keywords: statistical convergence; sequences of function; Cesàro summability

1 Introduction

The idea of statistical convergence was given by Zygmund [] in the first edition of his monograph published in Warsaw in . The concept of statistical convergence was in-troduced by Steinhaus [] and Fast [] and then reinin-troduced by Schoenberg [] inde-pendently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure the-ory, trigonometric series, turnpike theory and Banach spaces. Later on, it was further in-vestigated from the sequence space point of view and linked with summability theory by Connor [], Edelyet al.[], Etet al.[–], Fridy [], Güngöret al.[–], Kolk [], Orhanet al.[, ], Mursaleen [], Kumar and Mursaleen [], Rath and Tripathy [], Salat [], Savaş [] and many others. In recent years, generalizations of statistical conver-gence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.

In the present paper, we introduce and examine the concepts of pointwiseλ-statistical convergence of orderαand pointwise [V,λ]-summability of orderαof sequences of real valued functions. In Section , we give a brief overview about statistical convergence and strongp-Cesàro summability. In Section , we establish some inclusion relations between

wβ_λ_p(f) andSα

λ(f) and betweenS α

λ(f) andSλ(f).

2 Deﬁnition and preliminaries

The definitions of statistical convergence and strongp-Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and fol-lowed different lines of development since their first appearance. It turns out, however,

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that the two deﬁnitions can be simply related to one another in general and are equiva-lent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the setNof natural numbers. The density of a subsetEofNis deﬁned by

δ(E) =lim

n→∞



n n

k=

χE(k) provided the limit exists,

whereχEis the characteristic function ofE. It is clear that any ﬁnite subset ofNhas zero

natural density andδ(Ec) =  –δ(E).

Theα-densityof a subsetEofNwas deﬁned by Çolak []. Letαbe a real number such that  <α≤. Theα-densityof a subsetEofNis deﬁned by

δα(E) =lim

n



nα{k≤n:k∈E} provided the limit exists,

where|{k≤n:k∈E}|denotes the number of elements ofEnot exceedingn.

It is clear that any ﬁnite subset ofNhas zeroαdensity andδα(Ec) =  –δα(E) does not

hold for  <α<  in general, the equality holds only ifα= . Note that theα-densityof any set reduces to the natural density of the set in caseα= .

The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [] and after then statistical convergence of orderα and strongp-Cesàro summability of orderαstudied by Çolak [, ] and generalized by Çolak and Asma []. Letλ= (λn) be a nondecreasing sequence of positive real numbers tending to∞such

thatλn+≤λn+ ,λ= . The generalized de la Vallée-Poussin mean is deﬁned bytn(x) =



λn

k∈Inxk, whereIn= [n–λn+ ,n] forn= , , . . . . A sequencex= (xk) is said to be (V,

λ)-summable to a numberiftn(x)→asn→ ∞[]. Ifλn=n, then (V,λ)-summability is

reduced to Cesàro summability. Bywe denote the class of all nondecreasing sequence of positive real numbers tending to∞such thatλn+≤λn+ ,λ= .

Throughout the paper, unless stated otherwise, by ‘for alln∈Nno’ we mean ‘for alln∈ Nexcept ﬁnite numbers of positive integers’ whereNno={no,no+ ,no+ , . . .}for some no∈N={, , , . . .}.

LetAbe any non empty set, by B(A) we denote the class of all bounded real valued functions deﬁned onA.

3 Main results

In this section we give the main results of this paper. In Theorem ., we give the inclu-sion relations between the sets ofSα

λ(f)-statistically convergent sequences for diﬀerentαs

andμs. In Theorem ., we give the relationship between the strongwα

λp(f)-summability

and the strongwβμp(f)-summability. In Theorem ., we give the relationship between the

strongwβμp(f)-summability andSαλ(f)-statistical convergence.

Deﬁnition . Let the sequenceλ= (λn) be as above andα∈(, ] be any real number.

A sequence of functions{fk}is said to beSαλ(f)-statistical convergence (or pointwise

λ-statistically convergent of orderα) to the functionf on a setAif, for everyε> ,

lim

n

 λα

n

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where In= [n–λn+ ,n] andλαn denote theαth power (λn)α ofλn, that is λα= (λαn) =

(λα

,λα, . . . ,λαn, . . .). In this case, we write Sαλ–limfk(x) =f(x) onA. Sαλ–limfk(x) =f(x)

means that for everyδ>  and  <α≤, there is an integerNsuch that

 λα

n

k∈In:fk(x) –f(x)≥εfor everyx∈A<δ,

for alln>N(=N(ε,δ,x)) and for eachε> . The set of all pointwiseλ-statistically con-vergent function sequences of order α will be denoted by Sα

λ(f). In this case, we write

Sα

λ–limfk(x) =f(x) onA. Forλn=nfor alln∈N, we shall writeS

α₍_f_{) instead of}_Sα λ(f) and

in the special caseα= , we shall writeSλ(f) instead ofSλα(f).

TheSαλ(f)-statistical convergence is well deﬁned for  <α≤, but it is not well deﬁned

forα>  in general. Let us deﬁne the sequence{fk}as follows:

fk(x) = ⎧ ⎨ ⎩

, ifk= n, 

+kx ifk= n,

n= , , , . . .

then both

lim

n→∞  λα

n

_k_∈_I_n_:_f_k₍_x_{) – }_≥εfor everyx∈A≤ lim

n→∞ [λn] + 

λα

n

= 

and

lim

n→∞  λα

n k∈In:

fk(x) –

  +kx

≥εfor everyx∈A≤ lim

n→∞

([λn] + )

λα

n

= 

forα> , and soSα

λ(f)-statistically converges both to  and ,i.e. Sαλ–limfk(x) =  and Sαλ–limfk(x) = . But this is impossible.

Deﬁnition . Let the sequenceλ= (λn) be as above,α∈(, ] be any real number and

letpbe a positive real number. A sequence of functions{fk}is said to be stronglywαλp(f

)-summable (or pointwise [V,λ]-summable of orderα), if there is a functionf such that

lim

n→∞  λα

n

k∈In x∈A

fk(x) –f(x) p

= .

In this case, we writewαλp–limfk(x) =f(x) onA. The set of all stronglywβλp(f)-summable

sequences of function will be denoted bywα

λp(f). Forλn=nfor alln∈N, we shall write wα

p(f) instead ofwαλp(f) and in the special caseα= , we shall writewλp(f) instead ofwαλp(f).

Theorem . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nno,  <α≤β≤and{fk}be a sequence of real valued functions deﬁned on a set A.

(i) If

lim inf

n→∞

λα_n μβn

> 

thenSβ

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(ii) If

lim

n→∞

μn

λβn

= 

thenSα

λ(f)⊆Sβμ(f).

Proof (i) Suppose thatλn≤μnfor alln∈Nno and let () be satisﬁed. ThenIn⊂Jnand so

thatε>  we may write

k∈Jn:fk(x) –f(x)≥εfor everyx∈A

⊃k∈In:fk(x) –f(x)≥εfor everyx∈A

and so



μβn

≥ λαn

μβn

 λα

n

k∈In:fk(x) –f(x)≥εfor everyx∈A

for alln∈Nno, whereJn= [n–μn+ ,n]. Now taking the limit asn→ ∞in the last

in-equality and using (), we getSβ

μ(f)⊆Sλα(f).

(ii) LetSα

λ–limfk(x) =f(x) onAand () be satisﬁed. SinceIn⊂Jn, forε> , we may write

 μβn

=  μβn

_n_–μn+ ≤k≤n–λn:fk(x) –f(x)≥εfor everyx∈A

+  μβn

≤μn–λn

μβn

+  μβn

≤μn–λβn

λβn

+  μβn

≤

μn

λβn

– 

+  λα

n

for alln∈Nno. Sincelimn

μn

λβn

=  by () the ﬁrst term and sinceSα

λ–limfk(x) =f(x) onA,

the second term of right-hand side of above inequality tends to  asn→ ∞. (Note that (μn

λβn

– )≥ for alln∈Nno.) This implies thatS

α

λ(f)⊆Sμβ(f).

From Theorem ., we have the following results.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nnoand{fk}be a sequence of real valued functions deﬁned on a set A.If()holds then,

(i) Sα μ(f)⊆S

α

λ(f),for eachα∈(, ]and for allx∈A;

(ii) Sμ(f)⊆Sαλ(f),for eachα∈(, ]and for allx∈A;

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Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nnoand{fk}be a sequence of real valued functions deﬁned on a set A.If()holds then,

(i) Sα

λ(f)⊆Sαμ(f)for eachα∈(, ]and for allx∈A;

(ii) Sα

λ(f)⊆Sμ(f),for eachα∈(, ]and for allx∈A;

(iii) Sλ(f)⊆Sμ(f)for allx∈A.

Theorem . Given forλ= (λn),μ= (μn)∈suppose thatλn≤μnfor all n∈Nno,  <

α≤β≤and{fk}be a sequence of real valued functions deﬁned on a set A.Then (i) If()holds thenwβμp(f)⊂wαλp(f)for allx∈A;

(ii) If()holds and letf(x)∈B(A),thenB(A)∩wα λp(f)⊂w

β

μp(f)for allx∈A. Proof (i) Omitted.

(ii) Let (fk(x))∈B(A)∩wλαp(f) and suppose that () holds. Since (fk(x))∈B(A), then there

exists someM>  such that|fk(x) –f(x)| ≤Mfor allk∈Nand for allx∈A. Now, since

λn≤μnandIn⊂Jnfor alln∈Nno, we may write



μβn

k∈Jn x∈A

fk(x) –f(x) p

=  μβn

k∈Jn–In x∈A

fk(x) –f(x) p

+  μβn

k∈In x∈A

fk(x) –f(x) p

≤

μn–λn

μβn

Mp+  μβn

k∈In x∈A

fk(x) –f(x)p

≤

μn–λβn

μβn

Mp+  μβn

k∈In x∈A

_f_k(x) –f(x)p

≤

μn

λβn

– 

Mp+ 

λα

n

k∈In x∈A

fk(x) –f(x) p

for everyn∈Nno. Therefore,B(A)∩w

α λp(f)⊂w

β

μp(f).

From Theorem ., we have the following results.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nnoand{fk}be a sequence of real valued functions deﬁned on a set A.If()holds then:

(i) wα

μp(f)⊂wαλp(f),for eachα∈(, ]and for allx∈A; (ii) wμp(f)⊂wαλp(f),for eachα∈(, ]and for allx∈A; (iii) wμp(f)⊂wλp(f)for allx∈A.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nnoand{fk}be a sequence of real valued functions deﬁned on a set A.If()holds then:

(i) B(A)∩wα

λp(f)⊂wαμp(f),for eachα∈(, ]and for allx∈A; (ii) B(A)∩wα

λp(f)⊂wμp(f),for eachα∈(, ]and for allx∈A; (iii) B(A)∩wλp(f)⊂wμp(f)for allx∈A.

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(i) Let()holds,if a sequence of real valued functions deﬁned on a setAis strongly

wβμp(f)-summable tof,then it isSαλ(f)-statistically convergent tof;

(ii) Let()holds,f(x)∈B(A)and{fk}be a sequence of bounded real valued functions

deﬁned on a setA,if a sequence isSα

λ(f)-statistically convergent tof then it is strongly

wβμp(f)-summable tof.

Proof (i) For any function sequence (fk(x)) andε> , we have

k∈Jn x∈A

fk(x) –f(x) p

=

k∈Jn,x∈A

|fk(x)–f(x)|≥ε

fk(x) –f(x) p

+

k∈Jn,x∈A

|fk(x)–f(x)|<ε

fk(x) –f(x) p

≥

k∈In,x∈A

|fk(x)–f(x)|≥ε

fk(x) –f(x) p

+

k∈In,x∈A

|fk(x)–f(x)|<ε

fk(x) –f(x) p

≥

k∈In,x∈A

|fk(x)–f(x)|≥ε

fk(x) –f(x) p

≥k∈In:fk(x) –f(x)≥εfor everyx∈Aεp

and so that



μβn

k∈Jn x∈A

fk(x) –f(x) p

≥ 

μβn

k∈In:fk(x) –f(x)≥εfor everyx∈Aεp

≥ λαn

μβn

 λα

n

k∈In:fk(x) –f(x)≥εfor everyx∈Aεp.

Since () holds, it follows that if{fk}is stronglywβμp(f)-summable tof, then it isSαλ(f

)-statistically convergent tof. (ii) Suppose thatSα

λ–limfk(x) =f(x) and (fk(x))∈B(A). Then there exists someM> 

such that|fk(x) –f(x)| ≤Mfor allk, then for everyε>  we may write



μβn

k∈Jn x∈A

_f_k(x) –f(x)p =  μβn

k∈Jn–In x∈A

_f_k(x) –f(x)p+  μβn

k∈In x∈A

_f_k(x) –f(x)p

≤

μn–λn

μβn

Mp+  μβn

k∈In x∈A

fk(x) –f(x) p

≤

μn–λβn

μβn

Mp+  μβn

k∈In x∈A

fk(x) –f(x) p

=

μn

λβn

–λ

β

n

λβn

Mp+  μβn

k∈In,x∈A

|fk(x)–f(x)|≥ε

fk(x) –f(x)p

+  μβn

k∈In,x∈A |fk(x)–f(x)|<ε

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≤

μn

λβn

– 

Mp

+M

p

λα

n

k∈In:fk(x) –f(x)≥εfor everyx∈A+ε

for alln∈Nno. Using (), we obtain thatw

β

μp–limfk(x) =f(x), wheneverSαλ–limfk(x) =

f(x).

From Theorem ., we have the following results.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nnoandα∈(, ]be any real number.If()holds,then:

(i) If a sequence of real valued functions deﬁned on a setAis stronglywα

μp(f)-summable

tof,then it isSα

λ(f)-statistically convergent tof;

(ii) If a sequence of real valued functions deﬁned on a setAis stronglywμp(f)-summable

tof,then it isSα

λ(f)-statistically convergent tof;

(iii) If a sequence of real valued functions deﬁned on a setAis stronglywμp(f)-summable

tof,then it isSλ(f)-statistically convergent tof.

Corollary . Letλ= (λn)andμ= (μn)be two sequences insuch thatλn≤μnfor all n∈Nno,α∈(, ]be any real number.If()holds,then:

(i) If a sequence of bounded real valued functions deﬁned on a setAis

Sα

λ(f)-statistically convergent tof,then it is stronglywαμp(f)-summable tof; (ii) If a sequence of bounded real valued functions deﬁned on a set isSα

λ(f)-statistically

convergent tof,then it is stronglywμp(f)-summable tof;

(iii) If a sequence of bounded real valued functions deﬁned on a setAis

Sλ(f)-statistically convergent tof,then it is stronglywμp(f)-summable tof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ME, MÇ and MK have contributed to all parts of the article.

Author details

1_{Department of Mathematics, Fırat University, Elaz ˘g, 23119, Turkey.}2_{Department of Mathematics, Mu¸sAlparslan} University, Mu¸s, Turkey. 3_{Department of Mathematics, Bitlis Eren University, Bitlis, Turkey.}

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the Management Union of the Scientiﬁc Research Projects of Frat University for its ﬁnancial support under a grant with numberFUBAP FF.12.03. Also, the authors wish to thank the referees for their careful reading of the manuscript and valuable of the manuscript and valuable suggestions.

Received: 12 December 2012 Accepted: 10 April 2013 Published: 23 April 2013 References

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doi:10.1186/1029-242X-2013-204