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We have generalized the notion of statistical boundedness by introducing the concept of

The idea of statistical convergence was given in the first edition (published in Warsaw in 1935) of the monograph of Zygmund [

The standard definition of “

The number sequence

We will be particularly concerned with those subsets of

Using this notation, we have the following.

The number sequence

A number sequence

In 1997, Fridy and Orhan [

The number sequence

In the same year, that is, 1997, Tripathy [

Quite recently, Bhardwaj and Gupta [

The idea of a modulus function was structured by Nakano [

Connor [

In the year 2014, Aizpuru et al. [

Quite recently, Bhardwaj and Dhawan have introduced and studied the concepts of

Throughout the paper, unless otherwise specified,

First we recall some definitions from [

Let

For any unbounded modulus

A sequence

In view of Definition

A sequence

We now introduce the following notation: if

Using this notation, the definitions of

An

An

The main object of this paper is to introduce and study a new concept of

A number sequence

In this paper, we establish a relation between statistical boundedness and

Throughout this section, unless otherwise specified, we deal with

We begin this section by establishing a decomposition theorem for

If

As

For

If we denote the space of all

Fridy [

A sequence

First suppose

As an immediate consequence of Theorem

If

Aizpuru et al. [

Let

There exists a convergent sequence

We know that every subsequence of a convergent sequence is convergent but this is no longer true in case of

Consider

A subsequence of a sequence

Every

Burgin and Duman [

A sequence

The proof is similar to Theorem

The following theorem shows that continuous functions preserve the

If

As

Šalát [

The set

The proof is similar to Theorem

The above theorem provides us with the following information related to the structure of the set

The set

Since the sequence

In this section we show that the concept of

Throughout this section, we deal with the sequences of scalars.

Every bounded sequence is

The result follows in view of the fact that empty set has zero

Every

The proof follows in view of the fact that

The converse of the above theorem need not be true which can be verified by the following example.

Let

From Theorems

Aizpuru et al. [

A sequence

First, we suppose

Let

It is easy to note that

A subsequence of an

We now characterize

A sequence is

The proof is easy in view of Theorem

Every

The proof follows from the fact that

The following theorem shows that every

A sequence

The proof is easy and hence omitted.

Every

(a)

(b)

(c)

Every monotone and

Let

If

The following theorem shows that a sequence which is

If, for every unbounded modulus

Suppose, if possible,

From Theorem

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