Abstract

We introduce the concept of statistical convergence of order of difference sequences, and we give some relations between the set of statistical convergence of order of difference sequences and strong -summability of order . Furthermore some relations between the spaces and are examined.

1. Introduction

The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor [5], Fridy [6], Khan and Orhan [7], Miller and Orhan [8], Mursaleen [9], Rath and Tripathy [10], Šalát [11], Savaş [12], and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.

Let be the set of all sequences of real or complex numbers, and let , , and be, respectively, the Banach spaces of bounded, convergent, and null sequences with the usual norm , where , the set of positive integers. Also by , , , and , we denote the spaces of all bounded, convergent, absolutely, and -absolutely convergent series, respectively.

Let be a nondecreasing sequence of positive numbers tending to such that , . The generalized de la Vallée-Poussin mean is defined by , where for . A sequence is said to be -summable to a number if as [13]. If , then -summability is reduced to the Cesàro summability.

The notion of difference sequence spaces was introduced by Kızmaz [14], and it was generalized by Et and Çolak [15] such as for , , or , where , , , and so . The sequence spaces are the Banach spaces normed by for , or . Recently the difference sequence spaces have been studied in [16–21].

The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan [22] and after that statistical convergence of order and strongly -Cesàro summability of order were studied by Çolak [23] and generalized by Çolak and Bektaş [24].

Let be given. The sequence is said to be statistically convergent of order if there is a real number such that for every . In this case we write . The set of all statistically convergent sequences of order will be denoted by .

Let be a sequence space. Then is called(i)solid (or normal) if whenever for all sequences of scalars with ,(ii)symmetric if implies , where is a permutation of ,(iii)sequence algebra if is closed under multiplication.

2. Main Results

In this section we give the main results of the paper. In Theorem 5 we give the inclusion relations between the sets of -statistically convergent sequences of order for different ’s. In Theorem 9 we give the relationship between the strong -summability of order and the strong -summability of order . In Theorem 12 we give the relationship between the strong -summability of order and the -statistical convergence of order .

Definition 1. Let the sequence of real numbers be defined as above, and let be given. The sequence is said to be -statistically convergent of order if there is a real number such that where and denote the th power of ; that is . In this case we write . The set of all -statistically convergent sequences of order will be denoted by . For for all , we will write instead of and in the special case , we will write instead of , and also in the special case and for all we will write instead of , which were defined and studied by Et and Nuray [25].

The -statistical convergence of order is well defined for , but it is not well defined for in general. For this let for all and be defined as follows: and so then both for , and so -statistically converges of order , both to and ; that is, and . But this is impossible.

Theorem 2. Let and , be sequences of real numbers; then(i)if and , then ,(ii)if and , then .

Proof. Omitted.

Definition 3. Let the sequence be as above, any real number, and a positive real number. A sequence is said to be strongly -summable of order if there is a real number such that where . In this case we write . The strong -summability of order reduces to the strong -summability for . The set of all strongly -summable sequences of order will be denoted by .

Theorem 4. Let the sequence be as above, , and a positive real number. The sequence space is a Banach space for normed by and a complete -normed space for by

Proof. It is a routine verification that is a normed space normed by (9). Let be a Cauchy sequence such that for each . Then we have as and so that as . Now for we get as , and for all we get as . Since the inequality holds for all , we get as and for each . This implies that the sequence is a Cauchy in for each . Also, it is convergent, because is complete. Assume that for each . Since is a Cauchy sequence, then for every there exists a number such that for all . Hence for all and all we get Taking limit as in the last inequalities, we get for and all . This implies that and hence as . Since we get .
It can be shown that is a complete -normed space for by (10).

Theorem 5. If , then and the inclusion is strict.

Proof. Proof follows from the inequality To show that the inclusion is strict, define such that Then for , but for .

From Theorem 5 we have the following.

Corollary 6. If a sequence is -statistically convergent of order , to , then it is -statistically convergent to .

Theorem 7. if

Proof. For given we have and so Taking the limit as and using (21), we get .

Theorem 8. if and only if

Proof. From Theorem 7 we have .
Conversely, suppose that . We can choose a subsequence such that . If we consider a sequence defined by then but . From Corollary 6 we have . This completes the proof.

Theorem 9. Let and a positive real number; then and the inclusion is strict.

Proof. Let . Then given and such that and a positive real number , we may write and this gives that .
To show that the inclusion is strict define such that Then for but for , for .

The following result is a consequence of Theorem 9.

Corollary 10. Let and a positive real number. Then(i)if , then ,(ii) for each and .

Theorem 11. Let and . Then .

Proof is seen from the Hölder inequality.

Theorem 12. Let and be fixed real numbers such that and . If a sequence is strongly -summable of order , to , then it is -statistically convergent of order , to .

Proof. For any sequence and , we have and so From this it follows that if is strongly -summable of order , to , then it is -statistically convergent of order , to .

Corollary 13. Let be a fixed real number such that and .(i)If a sequence is strongly -summable of order , to , then it is -statistically convergent of order , to .(ii)If a sequence is strongly -summable of order , to , then it is -statistically convergent to .

3. Results Related to the Orlicz Function

In this section we give the inclusion relations between the sets of -statistically convergent sequences of order and strongly -summable sequences of order with respect to the Orlicz function .

An Orlicz function is a function , which is continuous, nondecreasing, and convex with , for and as . An Orlicz function can always be represented in the following integral form: where known as the kernel of is right differentiable for , , for , is nondecreasing, and as .

Let be a finite measure space. We denote by the space of all (equivalence classes of) -measurable functions from into . Given an Orlicz function , we define on a convex functional by and an Orlicz space by (for detail see [26]).

The study of Orlicz sequence spaces was initiated with a certain specific purpose in the Banach space theory. Indeed, Lindberg [27] got interested in Orlicz spaces in connection with finding the Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to or . Subsequently, Lindenstrauss and Tzafriri [28] used the idea of the Orlicz function to construct the sequence space The space with the norm becomes a Banach space, called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with for . Lindenstrauss and Tzafriri [28] proved that every Orlicz sequence space contains a subspace isomorphic to   . The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [26].

It is well known that if is a convex function and , then for all with .

Recently, the Orlicz sequence spaces have been studied by Altin et al. [29], Bhardwaj and Singh [30], Et et al. [31, 32], Mursaleen et al. [33], Savaş and Rhoades [34], and many others.

Definition 14. Let an Orlicz function, a sequence of strictly positive real numbers, and any real number. Now we define If , then we say that is strongly -summable of order with respect to the Orlicz function . For for all and we will write instead of and in the special case we will write instead of .