International Journal of Analysis
Volume 2016 (2016), Article ID 7283527, 4 pages
http://dx.doi.org/10.1155/2016/7283527

Research Article

On Inclusion Relations between Some Sequence Spaces

R. Çolak, Ç. A. Bektaş, H. Altınok, and S. Ercan

Department of Mathematics, Firat University, 23119 Elâzığ, Turkey

Received 15 April 2016; Accepted 10 July 2016

Academic Editor: Julien Salomon

Copyright © 2016 R. Çolak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We determine the relations between the classes of almost -statistically convergent sequences and the relations between the classes of strongly almost -summable sequences for various sequences , in the class . Furthermore we also give the relations between the classes of almost -statistically convergent sequences and the classes of strongly almost -summable sequences for various sequences .

1. Introduction

A sequence of real (or complex) numbers is said to be statistically convergent to the number if for every In this case, we write or and denotes the set of all statistically convergent sequences.

A sequence of real (or complex) numbers is said to be almost statistically convergent to the number if for every In this case, we write or and denotes the set of all almost statistically convergent sequences [1].

Let be a nondecreasing sequence of positive real numbers tending to such that The set of all such sequences will be denoted by

The generalized de la Vallée-Poussin mean is defined by where .

A sequence is said to be -summable to a number (see [2]) if If for each , then -summability reduces to -summability.

We write for the sets of sequences which are strongly Cesàro summable and strongly -summable to ; that is, and , respectively.

Savaš [1] defined the following sequence space: for the sets of sequences which are strongly almost -summable to ; that is, . We will write

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

Let . A sequence is said to be -statistically convergent or -convergent to if for every where . In this case we write or , and .

The sequence is said to be -almost statistically convergent if there is a complex number such that In this case, we write or and denotes the set of all -almost statistically convergent sequences. If we choose for all , then -almost statistical convergence reduces to almost statistical convergence [1].

2. Main Results

Throughout the paper, unless stated otherwise, by “for all ” we mean “for all except finite numbers of positive integers” where for some .

Theorem 1. Let and be two sequences in such that for all Consider the following:  (i)If then . (ii)If then .

Proof. (i) Suppose that for all and let (10) be satisfied. Then so that for we may writeand therefore we have for all , where Now taking the limit as uniformly in in the last inequality and using (10) we get so that .
(ii) Let and (11) be satisfied. Since , for , we may write for all . Since by (11) and since the first term and second term of right hand side of above inequality tend to 0 as uniformly in . This implies that as uniformly in Therefore .

From Theorem 1 we have the following result.

Corollary 2. Let and be two sequences in such that for all . If (11) holds then .

If we take in Corollary 2 we have the following result.

Corollary 3. Let If then we have

Theorem 4. Let and and suppose that for all . Consider the following: (i)If (10) holds then .(ii)If (11) holds then .

Proof. (i) Suppose that for all . Then so that we may write for all . This gives that Then taking limit as , uniformly in in the last inequality, and using (10) we obtain .
(ii) Let be any sequence. Suppose that and that (11) holds. Since then there exists some such that for all and . Since so that , and for all , we may write for every . Since by (11) and since the first term and the second term of right hand side of above inequality tend to 0 as , uniformly in (note that for all ). Then we get . Since is an arbitrary sequence we obtain .

Since clearly (11) implies (10) from Theorem 4 we have the following result.

Corollary 5. Let such that for all . If (11) holds then .

Theorem 6. Let such that for all . Consider the following:  (i)If (10) holds then  and the inclusion holds for some . (ii)If and then , whenever (11) holds. (iii)If (11) holds then .

Proof. (i) Let be given and let . Now for every we may write so that for all . Then taking limit as , uniformly in in the last inequality, and using (10) we obtain that whenever . Since is an arbitrary sequence we obtain that .
(ii) Suppose that and . Then there exists some such that for all and Since , then for every we may writefor all Using (11) we obtain that whenever Since is an arbitrary sequence we obtain .
(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 then .

If we take for all in Theorem 6 then we have the following results, because implies that ; that is, (11) ⇒ (10).

Corollary 8. If then (i)if and then ,(ii)if then .

Competing Interests

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

References

E. Savaš, “Strong almost convergence and almost -statistical convergence,” Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531–536, 2000.

L. Leindler, “Über die de la vallée-pousinsche summierbarkeit allgemeiner Orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965.

M. Mursaleen, “-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.