IJANAL International Journal of Analysis 2314-4998 2314-498X Hindawi Publishing Corporation 10.1155/2016/7283527 7283527 Research Article On Inclusion Relations between Some Sequence Spaces http://orcid.org/0000-0001-8161-5186 Çolak R. 1 http://orcid.org/0000-0003-0397-3193 Bektaş Ç. A. 1 Altınok H. 1 http://orcid.org/0000-0001-9871-2142 Ercan S. 1 Salomon Julien Department of Mathematics Firat University 23119 Elâzığ Turkey firat.edu.tr 2016 382016 2016 15 04 2016 10 07 2016 2016 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.

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 = x k of real (or complex) numbers is said to be statistically convergent to 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 sequence x = x k of real (or complex) numbers is said to be almost statistically convergent to 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 .

Let λ = 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 sequence x = x k is said to be V , λ -summable to a number L (see ) 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 L for 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  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   m for 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 .

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

Let λ = 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 sequence x = 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 .

2. Main Results

Throughout the paper, unless stated otherwise, by “for all n 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.

Let λ = 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:

If (10) liminf n λ n μ n > 0

then S ^ μ S ^ λ .

If (11) lim n λ n μ n = 1

then S ^ λ S ^ μ .

Proof.

(i) Suppose that λ 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 ε and 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 ε for 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) Let x 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 ε for 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 Theorem 1 we have the following result.

Corollary 2.

Let λ = 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 take μ = μ n = n in Corollary 2 we have the following result.

Corollary 3.

Let λ = λ n Λ . If lim n λ n / n = 1 then we have S ^ λ = S ^ .

Theorem 4.

Let λ = λ n and μ = μ n Λ and suppose that λ n μ n for all n N n o . Consider the following:

If (10) holds then V ^ , μ V ^ , λ .

If (11) holds then V ^ , λ V ^ , μ .

Proof.

(i) Suppose that λ 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) Let x = 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 - L for 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.

Let λ , μ Λ such that λ n μ n for all n N n o . If (11) holds then V ^ , λ = V ^ , μ .

Theorem 6.

Let λ , μ Λ such that λ n μ n for all n N n o . Consider the following:

If (10) holds then (18) x k L V ^ , μ x k L S ^ λ

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

If x k l and x k L S ^ λ then x k L V ^ , μ , whenever (11) holds.

If (11) holds then l S ^ λ = V ^ , μ .

Proof.

(i) Let ε > 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 ε for 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 that x 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 + ε for 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 Theorem 1(i) and Theorem 6(i) we obtain the following result.

Corollary 7.

If liminf n λ n / μ n > 0 then S ^ μ V ^ , μ S ^ λ .

If we take μ 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.

If lim n λ n / n = 1 then

if x k l and x k L S ^ λ then x k L C , 1 ,

if x k L C , 1 then x k L S ^ λ .

Competing Interests

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

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