# On Inclusion Relations between Some Sequence Spaces.

1. Introduction

A sequence x = ([x.sub.k]) of real (or complex) numbers is said to be statistically convergent to the number L if for every [epsilon] > 0

[mathematical expression not reproducible]. (1)

In this case, we write S - lim x = L or [x.sub.k] [right arrow] L(S) and S denotes the set of all statistically convergent sequences.

A sequence x = ([x.sub.k]) of real (or complex) numbers is said to be almost statistically convergent to the number L if for every [epsilon] > 0

[mathematical expression not reproducible]. (2)

In this case, we write [??]-lim x = L or [x.sub.k] [right arrow] L([??]) and [??] denotes the set of all almost statistically convergent sequences [1].

Let [lambda] = ([[lambda].sub.n]) be a nondecreasing sequence of positive real numbers tending to [infinity] such that

[[lambda].sub.n+1] [less than or equal to] [[lambda].sub.n] + 1, [[lambda].sub.1] = 1. (3)

The set of all such sequences will be denoted by A.

The generalized de la Vallee-Poussin mean is defined by

[t.sub.n] = 1/[[lambda].sub.n] [summation over (k[member of][I.sub.n])] [x.sub.k], (4)

where [I.sub.n] = [n - [[lambda].sub.n] + 1, n].

A sequence x = ([x.sub.k]) is said to be ([LAMBDA], [lambda])-summable to a number L (see [2]) if

[t.sub.n] (x) [right arrow] L as n [right arrow] [infinity]. (5)

If [[lambda].sub.n] = n for each n [member of] N, then (V, [lambda])-summability reduces to (C, 1)-summability.

We write

[mathematical expression not reproducible] (6)

for the sets of sequences x = ([x.sub.k]) which are strongly Cesaro summable and strongly (V, [lambda])-summable to L; that is, [x.sub.k] [right arrow] L[C, 1] and [x.sub.k] [right arrow] L[V, [lambda]], respectively.

Savas [1] defined the following sequence space:

[mathematical expression not reproducible] (7)

for the sets of sequences x = ([x.sub.k]) which are strongly almost (V, [lambda])-summable to L; that is, [x.sub.k] [right arrow] L[V, [lambda]]. We will write [[[??], [lambda]].sub.[infinity]] = [[??], [lambda]] [intersection] [l.sub.[infinity]] .

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

Let [lambda] = ([[lambda].sub.n]) [member of] [LAMBDA]. A sequence x = ([x.sub.k]) is said to be [lambda]-statistically convergent or [S.sub.[lambda]]-convergent to L if for every [epsilon] > 0

[mathematical expression not reproducible], (8)

where [I.sub.n] = [n - [[lambda].sub.n] + l, n]. In this case we write [S.sub.[lambda]]-lim x = L or [x.sub.k] [right arrow] L([S.sub.[lambda]]), and [S.sub.[lambda]] = {x = ([x.sub.k]) : [S.sub.[lambda]] - lim x = L for some L}.

The sequence x = ([x.sub.k]) is said to be [lambda]-almost statistically convergent if there is a complex number L such that

[mathematical expression not reproducible], (9)

uniformly in m.

In this case, we write [[??].sub.x] - lim x = L or [x.sub.k] [right arrow] L([[??].sub.X]) and [[??].sub.[lambda]] denotes the set of all [lambda]-almost statistically convergent sequences. If we choose [[lambda].sub.n] = n for all n, then [lambda]-almost statistical convergence reduces to almost statistical convergence [1].

2. Main Results

Throughout the paper, unless stated otherwise, by "for all [mathematical expression not reproducible]" we mean "for all n [member of] N except finite numbers of positive integers" where [mathematical expression not reproducible] for some [n.sub.o] [member of] N = {1, 2, 3, ...}.

Theorem 1. Let [lambda] = ([[lambda].sub.n]) and [mu] = ([[mu].sub.n]) be two sequences in [LAMBDA] such that [[lambda].sub.n] [less than or equal to] [[mu].sub.n] for all [mathematical expression not reproducible], Consider the following:

(i) If

[mathematical expression not reproducible] (10)

then [[??].sub.[mu]] [subset or equal to] [[??].sub.[lambda]].

(ii) If

[mathematical expression not reproducible] (11)

then [[??].sub.[lambda]] [subset or equal to] [[??].sub.[mu]].

Proof. (i) Suppose that [X.sub.n] [less than or equal to] [[lambda].sub.n] for all [mathematical expression not reproducible] and let (10) be satisfied. Then [I.sub.n] [subset] [J.sub.n] so that for [epsilon] > 0 we may write

[mathematical expression not reproducible] (12)

and therefore we have

[mathematical expression not reproducible] (13)

for all [mathematical expression not reproducible], where [J.sub.n] = [n - [[mu].sub.n] +1, n]. Now taking the limit as n [right arrow] [infinity] uniformly in m in the last inequality and using (10) we get [mathematical expression not reproducible] so that [[??].sub.[mu]] [subset or equal to] [[??].sub.[lambda]].

(ii) Let ([x.sub.k]) [member of] [[??].sub.[lambda]] and (11) be satisfied. Since [I.sub.n] [subset] [J.sub.n], for [epsilon] > 0, we may write

[mathematical expression not reproducible] (14)

for all [mathematical expression not reproducible]. Since [lim.sub.n]([[lambda].sub.n]/[[mu].sub.n]) = 1 by (11) and since x = ([x.sub.k]) [member of] [[??].sub.[lambda]] the first term and second term of right hand side of above inequality tend to O as n [right arrow] [infinity] uniformly in m. This implies that (1/[[mu].sub.n])[absolute value of ({k [member of] [J.sub.n] : [absolute value of ([x.sub.k+m] - L)] [greater than or equal to] [epsilon]})] [right arrow] 0 as n [right arrow] [infinity] uniformly in m. Therefore [[??].sub.[lambda]] [subset or equal to] [[??].sub.[mu]].

From Theorem 1 we have the following result.

Corollary 2. Let [lambda] = ([[lambda].sub.n]) and [mu] = ([[mu].sub.n]) be two sequences in [[lambda].sub.n] such that [[mu].sub.n] < for all [mathematical expression not reproducible]. If (11) holds then [[??].sub.[lambda]] = [[??].sub.[mu]].

If we take [mu] = ([[mu].sub.n]) = (n) in Corollary 2 we have the following result.

Corollary 3. Let [lambda] = ([[lambda].sub.n]) [member of] [LAMBDA]. If [lim.sub.n]([[lambda].sub.n]/n) = 1 then we have [[??].sub.[lambda]] = [??].

Theorem 4. Let [lambda] = ([[lambda].sub.n]) and [mu] = ([[mu].sub.n]) [member of] [LAMBDA] and suppose that [[lambda].sub.n] < [[mu].sub.n] for all [mathematical expression not reproducible]. Consider the following:

(i) If (10) holds then [[??], [mu]] [subset or equal to] [[??], [lambda]].

(ii) If (11) holds then [[[??], [lambda]].sub.[infinity]] [subset or equal to] [[??], [mu]].

Proof. (i) Suppose that [[lambda].sub.n] < for all [mathematical expression not reproducible]. Then [I.sub.n] [subset or equal to] [J.sub.n] so that we may write

[mathematical expression not reproducible] (15)

for all [mathematical expression not reproducible]. This gives that

[mathematical expression not reproducible]. (16)

Then taking limit as n [right arrow] [infinity], uniformly in m in the last inequality, and using (10) we obtain [mathematical expression not reproducible].

(ii) Let x = ([x.sub.k]) [member of] [[??], [[lambda]].sub.[infinity]] be any sequence. Suppose that [x.sub.k] [right arrow] L[[??], [mu]] and that (11) holds. Since x = ([x.sub.k]) [member of] [l.sub.[infinity]] then there exists some M > 0 such that [absolute value of ([x.sub.k+m] - L)] [less than or equal to] M for all k and m. Since [[lambda].sub.n] [less than or equal to] [[mu].sub.n] so that 1/[[mu].sub.n] [less than or equal to] 1/[[lambda].sub.n], and [I.sub.n] [subset] [J.sub.n] for all [mathematical expression not reproducible], we may write

[mathematical expression not reproducible] (17)

for every [mathematical expression not reproducible]. Since [lim.sub.n]([[lambda].sub.n]/[[lambda].sub.n]) = 1 by (11) and since [x.sub.k] [right arrow] L[[??], [lambda]] the first term and the second term of right hand side of above inequality tend to 0 as n [right arrow] [infinity], uniformly in m (note that 1 - [[lambda].sub.n]/[[mu].sub.n] [greater than or equal to] 0 for all [mathematical expression not reproducible]). Then we get [mathematical expression not reproducible]. Since x = ([x.sub.k]) [member of] [[[??], [lambda]].sub.[infinity]] is an arbitrary sequence we obtain [[[??], [lambda]].sub.[infinity]] [subset or equal to] [[??], [mu]].

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

Corollary 5. Let [lambda], [mu] [member of] [LAMBDA] such that [[lambda].sub.n] [less than or equal to] [[mu].sub.n] for all [mathematical expression not reproducible]. If (11) holds then [[[??], [lambda]].sub.[infinity]] = [[[??], [mu]].sub.[infinity]].

Theorem 6. Let [lambda], [mu] [member of] [LAMBDA] such that [[lambda].sub.n] [less than or equal to] [[mu].sub.n] for all [mathematical expression not reproducible]. Consider the following:

(i) If (10) holds then

[mathematical expression not reproducible] (18)

and the inclusion [mathematical expression not reproducible] holds for some [lambda], [mu] [member of] [LAMBDA].

(ii) If ([x.sub.k]) [member of] [l.sub.[infinity]] and [x.sub.k] [right arrow] L([[??].sub.[lambda]]) then [x.sub.k] [right arrow] L[[??], [mu]], whenever (11) holds.

(iii) If (11) holds then [l.sub.[infinity]] [intersection] [[??].sub.[lambda]] = [[[??], [mu]].sub.[infinity]]

Proof. (i) Let [epsilon] > 0 be given and let [x.sub.k] [right arrow] L[[??], [mu]]. Now for every [epsilon] > 0 we may write

[mathematical expression not reproducible] (19)

so that

[mathematical expression not reproducible] (20)

for all [mathematical expression not reproducible]. Then taking limit as n [right arrow] [infinity], uniformly in m in the last inequality, and using (10) we obtain that [x.sub.k] [right arrow] L([[??].sub.[lambda]]) whenever [x.sub.k] [right arrow] L[[??], [mu]]. Since x = ([x.sub.k]) [member of] [[??], [mu]] is an arbitrary sequence we obtain that [[??], [mu]] [subset] [[??].sub.[lambda]].

(ii) Suppose that [x.sub.k] [right arrow] L([[??].sub.[lambda]]) and x = ([x.sub.k]) [member of] [l.sub.[infinity]]. Then there exists some M > 0 such that [absolute value of ([x.sub.k+m] - L)] [less than or equal to] M for all k and m. Since 1/[[mu].sub.n] [less than or equal to] 1/[[lambda].sub.n], then for every [epsilon] > 0 we may write

[mathematical expression not reproducible] (21)

for all [mathematical expression not reproducible]. Using (11) we obtain that [x.sub.k] [right arrow] L[[??], [mu]] whenever [x.sub.k] [right arrow] L([[??].sub.[lambda]]). Since x = ([x.sub.k]) [member of] [l.sub.[infinity]] [intersection] [[??].sub.[lambda]] is an arbitrary sequence we obtain [l.sub.[infinity]] [intersection] [[??].sub.[lambda]] [subset or equal to] [[??], [mu]].

(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 lim [inf.sub.n[right arrow][infinity]] ([[lambda].sub.n]/[[mu].sub.n]) > 0 then [[??].sub.[mu]] [intersection] [[??], [mu]] [subset] [[??].sub.[lambda]].

If we take [[mu].sub.n] = n for all n in Theorem 6 then we have the following results, because [lim.sub.n[right arrow][infinity]] ([[lambda].sub.n]/[[lambda].sub.n]) = 1 implies that lim [inf.sub.n[right arrow][infinity]] ([[lambda].sub.n]/[[mu].sub.n]) = 1 > 0; that is, (11) [??] (10).

Corollary 8. If [lim.sub.n[right arrow][infinity]] ([[lambda].sub.n]/n) = 1 then

(i) if ([x.sub.k]) [member of] [l.sub.[infinity]] and [x.sub.k] [right arrow] L([[??].sub.[lambda]]) then [x.sub.k] [right arrow] L[C, 1],

(ii) if [x.sub.k] [right arrow] L[C, 1] then [x.sub.k] [right arrow] L([[??].sub.[lambda]]).

http://dx.doi.org/10.1155/2016/7283527

Competing Interests

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

References

[1] E. Savas, "Strong almost convergence and almost A-statistical convergence," Hokkaido Mathematical Journal, vol. 29, no. 3, pp. 531-536, 2000.

[2] L. Leindler, "Uber die de la vallee-pousinsche summierbarkeit allgemeiner Orthogonalreihen," Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375-387, 1965.

[3] M. Mursaleen, "A-statistical convergence," Mathematica Slovaca, vol. 50, no. 1, pp. 111-115, 2000.

R. Colak, A. Bektas, H. Altinok, and S. Ercan

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Correspondence should be addressed to C. A. Bektas; cbektas@firat.edu.tr

Received 15 April 2016; Accepted 10 July 2016