- Naim L Braha
^{1}Email author

**2011**:539745

https://doi.org/10.1155/2011/539745

© Naim L. Braha. 2011

**Received: **5 November 2010

**Accepted: **18 February 2011

**Published: **10 March 2011

## Abstract

We introduce new sequence space defined by combining an Orlicz function, seminorms, and -sequences. We study its different properties and obtain some inclusion relation involving the space Inclusion relation between statistical convergent sequence spaces and Cesaro statistical convergent sequence spaces is also given.

## 1. Introduction

By , we denote the space of all real or complex valued sequences. If , then we simply write instead of . Also, we will use the conventions that . Any vector subspace of is called a sequence space. We will write , , and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by , we denote the sequence space of all -absolutely convergent series, that is, for . Throughout the article, , , and denote, respectively, the spaces of all, bounded, and -absolutely summable sequences with the elements in , where is a seminormed space. By , we denote the zero element in . denotes the set of all subsets of , that do not contain more than elements. With , we will denote a nondecreasing sequence of positive real numbers such that and , as . The class of all the sequences satisfying this property is denoted by .

which yields that and hence is -convergent to . We therefore deduce that the ordinary convergence implies the -convergence to the same limit.

## 2. Definitions and Background

The space is closely related to the space which is an Orlicz sequence space with . An Orlicz function is a function which is continuous, nondecreasing, and convex with , for and as . It is well known that if is a convex function and , then for all with .

An Orlicz function is said to satisfy the -condition for all values of , if there exists a constant such that (see, Krasnoselskii and Rutitsky [8]). In the later stage, different Orlicz sequence spaces were introduced and studied by Bhardwaj and Singh [9], Güngör et al. [10], Tripathy and Mahanta [6], Esi [11], Esi and Et [12], Parashar and Choudhary [13], and many others.

## 3. Results

Since the proofs of the following theorems are not hard we omit them.

Theorem 3.1.

The sequence spaces are linear spaces over the complex field .

Theorem 3.2.

In what follows, we will show inclusion theorems between spaces .

Theorem 3.3.

Proof.

Therefore, , which is contradiction.

Corollary 3.4.

Theorem 3.5.

Let , , be Orlicz functions which satisfy the -condition and , and seminorms. Then

(4)If is stronger than , then , and

(5)If is equivalent to , then .

Proof.

Proof is similar to [14, Theorem 2.5].

Corollary 3.6.

Let be an Orlicz function which satisfy the -condition. Then .

Theorem 3.7.

Let be a sequence of Orlicz functions. Then the sequence space is solid and monotone.

Proof.

## 4. Statistical Convergence

Theorem 4.1.

If is any Orlicz function, strictly increasing sequence, then .

Proof.

Taking the limit as , it follows that .

Theorem 4.2.

If is any Orlicz bounded function, strictly increasing sequence, then , for every .

Proof.

where the summation is over and the summation is over . Taking the limit as and , it follows that .

## 5. Cesaro Convergence

Theorem 5.1.

If is an Orlicz function. Then .

Proof.

Theorem 5.2.

Let . Then for any Orlicz function, , .

Proof.

Theorem 5.3.

## Authors’ Affiliations

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