#### 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.