We determine the relations between the classes S^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.

1. Introduction

A sequence x=xk of real (or complex) numbers is said to be statistically convergent to the number L if for every ε>0(1)limn→∞1nk≤n:xk-L≥ε=0.In this case, we write S-limx=L or xk→LS and S denotes the set of all statistically convergent sequences.

A sequence x=xk of real (or complex) numbers is said to be almost statistically convergent to the number L if for every ε>0(2)limn→∞1nk≤n:xk+m-L≥ε=0uniformly in m.In this case, we write S^-limx=L or xk→LS^ and S^ denotes the set of all almost statistically convergent sequences [1].

Let λ=λn be a nondecreasing sequence of positive real numbers tending to ∞ such that (3)λn+1≤λn+1,λ1=1.The set of all such sequences will be denoted by Λ.

The generalized de la Vallée-Poussin mean is defined by (4)tnx=1λn∑k∈Inxk, where In=n-λn+1,n.

A sequence x=xk is said to be V,λ-summable to a number L (see [2]) if (5)tnx⟶Las n⟶∞. If λn=n for each n∈N, then V,λ-summability reduces to C,1-summability.

We write (6)C,1=x=xk:limn→∞1n∑k=1nxk-L=0 for some L,V,λ=x=xk:limn→∞1λn∑k∈Inxk-L=0 for some Lfor the sets of sequences x=xk which are strongly Cesàro summable and strongly V,λ-summable to L; that is, xk→LC,1 and xk→LV,λ, respectively.

Savaš [1] defined the following sequence space: (7)V^,λ=x=xk: limn→∞1λn∑k∈Inxk+m-L=0 for some L, uniformly in mfor the sets of sequences x=xk which are strongly almost V,λ-summable to L; that is, xk→LV,λ. We will write V^,λ∞=V^,λ∩l∞.

The λ-statistical convergence was introduced by Mursaleen in [3] as follows.

Let λ=λn∈Λ. A sequence x=xk is said to be λ-statistically convergent or Sλ-convergent to L if for every ε>0(8)limn→∞1λnk∈In:xk-L≥ε=0,where In=n-λn+1,n. In this case we write Sλ-limx=L or xk→L(Sλ), and Sλ=x=xk:Sλ-limx=L for some L.

The sequence x=xk is said to be λ-almost statistically convergent if there is a complex number L such that (9)limn→∞1λnk∈In:xk+m-L≥ε=0,uniformly in m.In this case, we write S^λ-limx=L or xk→L(S^λ) and S^λ denotes the set of all λ-almost statistically convergent sequences. If we choose λn=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 all n∈Nno” we mean “for all n∈N except finite numbers of positive integers” where Nno=no,no+1,no+2,… for some no∈N=1,2,3,….

Theorem 1.

Let λ=λn and μ=μn be two sequences in Λ such that λn≤μn for all n∈Nno. Consider the following:

If(10)liminfn→∞λnμn>0

then S^μ⊆S^λ.

If(11)limn→∞λnμn=1

then S^λ⊆S^μ.

Proof.

(i) Suppose that λn≤μn for all n∈Nno and let (10) be satisfied. Then In⊂Jn so that for ε>0 we may write(12)k∈Jn:xk+m-L≥ε≥k∈In:xk+m-L≥εand therefore we have (13)1μnk∈Jn:xk+m-L≥ε≥λnμn1λnk∈In:xk+m-L≥εfor all n∈Nno, where Jn=n-μn+1,n. Now taking the limit as n→∞ uniformly in m in the last inequality and using (10) we get xk→LS^μ⇒xk→LS^λ so that S^μ⊆S^λ.

(ii) Let xk∈S^λ and (11) be satisfied. Since In⊂Jn, for ε>0, we may write (14)1μnk∈Jn:xk+m-L≥ε=1μnn-μn+1≤k≤n-λn:xk+m-L≥ε+1μnk∈In:xk+m-L≥ε≤μn-λnμn+1λnk∈In:xk+m-L≥ε≤1-λnμn+1λnk∈In:xk+m-L≥εfor all n∈Nno. Since limnλn/μn=1 by (11) and since x=xk∈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/μnk∈Jn:xk+m-L≥ε→0 as n→∞ uniformly in m. Therefore S^λ⊆S^μ.

From Theorem 1 we have the following result.

Corollary 2.

Let λ=λn and μ=μn be two sequences in Λ such that λn≤μn for all n∈Nno. If (11) holds then S^λ=S^μ.

If we take μ=μn=n in Corollary 2 we have the following result.

Corollary 3.

Let λ=λn∈Λ. If limnλn/n=1 then we have S^λ=S^.

Theorem 4.

Let λ=λn and μ=μn∈Λ and suppose that λn≤μn for all n∈Nno. Consider the following:

If (10) holds then V^,μ⊆V^,λ.

If (11) holds then V^,λ∞⊆V^,μ.

Proof.

(i) Suppose that λn≤μn for all n∈Nno. Then In⊆Jn so that we may write (15)1μn∑k∈Jnxk+m-L≥1μn∑k∈Inxk+m-L for all n∈Nno. This gives that (16)1μn∑k∈Jnxk+m-L≥λnμn1λn∑k∈Inxk+m-L.Then taking limit as n→∞, uniformly in m in the last inequality, and using (10) we obtain xk→LV^,μ⇒xk→LV^,λ.

(ii) Let x=xk∈V^,λ∞ be any sequence. Suppose that xk→LV^,μ and that (11) holds. Since x=xk∈l∞ then there exists some M>0 such that xk+m-L≤M for all k and m. Since λn≤μn so that 1/μn≤1/λn, and In⊂Jn for all n∈Nno, we may write (17)1μn∑k∈Jnxk+m-L=1μn∑k∈Jn-Inxk+m-L+1μn∑k∈Inxk+m-L≤μn-λnμnM+1μn∑k∈Inxk+m-L≤1-λnμnM+1λn∑k∈Inxk+m-Lfor every n∈Nno. Since limnλn/μn=1 by (11) and since xk→LV^,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-λn/μn≥0 for all n∈Nno). Then we get xk→LV^,λ⇒xk→LV^,μ. Since x=xk∈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.

Let λ,μ∈Λ such that λn≤μn for all n∈Nno. If (11) holds then V^,λ∞=V^,μ∞.

Theorem 6.

Let λ,μ∈Λ such that λn≤μn for all n∈Nno. Consider the following:

If (10) holds then (18)xk⟶LV^,μ⟹xk⟶LS^λ

and the inclusion V^,μ⊂S^λ holds for some λ,μ∈Λ.

If xk∈l∞ and xk→LS^λ then xk→LV^,μ, whenever (11) holds.

If (11) holds then l∞∩S^λ=V^,μ∞.

Proof.

(i) Let ε>0 be given and let xk→LV^,μ. Now for every ε>0 we may write (19)∑k∈Jnxk+m-L≥∑k∈Inxk+m-L≥∑k∈Inxk+m-L≥εxk+m-L≥εk∈In:xk+m-L≥ε so that (20)1μn∑k∈Jnxk+m-L≥λnμn1λnk∈In:xk+m-L≥εεfor all n∈Nno. Then taking limit as n→∞, uniformly in m in the last inequality, and using (10) we obtain that xk→LS^λ whenever xk→LV^,μ. Since x=xk∈V^,μ is an arbitrary sequence we obtain that V^,μ⊂S^λ.

(ii) Suppose that xk→LS^λ and x=xk∈l∞. Then there exists some M>0 such that xk+m-L≤M for all k and m. Since 1/μn≤1/λn, then for every ε>0 we may write(21)1μn∑k∈Jnxk+m-L=1μn∑k∈Jn-Inxk+m-L+1μn∑k∈Inxk+m-L≤μn-λnμnM+1μn∑k∈Inxk+m-L≤1-λnμnM+1λn∑k∈Inxk+m-L≤1-λnμnM+1λn∑k∈Inxk+m-L≥εxk+m-L+1λn∑k∈Inxk+m-L<εxk+m-L≤1-λnμnM+Mλnk∈In:xk+m-L≥ε+εfor all n∈Nno. Using (11) we obtain that xk→LV^,μ whenever xk→LS^λ. Since x=xk∈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 Theorem 1(i) and Theorem 6(i) we obtain the following result.

Corollary 7.

If liminfn→∞λn/μn>0 then S^μ∩V^,μ⊂S^λ.

If we take μn=n for all n in Theorem 6 then we have the following results, because limn→∞λn/μn=1 implies that liminfn→∞λn/μn=1>0; that is, (11) ⇒ (10).

Corollary 8.

If limn→∞λn/n=1 then

if xk∈l∞ and xk→LS^λ then xk→LC,1,

if xk→LC,1 then xk→LS^λ.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

SavašE.Strong almost convergence and almost λ-statistical convergenceLeindlerL.Über die de la vallée-pousinsche summierbarkeit allgemeiner OrthogonalreihenMursaleenM.λ-statistical convergence