- Research Article
- Open Access

# A New Class of Sequences Related to the Spaces Defined by Sequences of Orlicz Functions

- 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.

## Keywords

- Banach Space
- Natural Number
- Topological Space
- Sequence Space
- Positive Real Number

## 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

(1) ,

(2) ,

(3) ,

(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.

which proves that is solid space and monotone.

## 4. Statistical Convergence

for some .

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.

Hence .

Theorem 5.2.

Let . Then for any Orlicz function, , .

Proof.

from this, it follows that .

Theorem 5.3.

Let . Then for any Orlicz function, , .

## Authors’ Affiliations

## References

- Mursaleen M, Noman AK:
**On the spaces of**λ**-convergent and bounded sequences.***Thai Journal of Mathematics*2010,**8**(2):311–329.MathSciNetMATHGoogle Scholar - Sargent WLC:
**Some sequence spaces related to the****spaces.***Journal of the London Mathematical Society*1960,**35:**161–171. 10.1112/jlms/s1-35.2.161MathSciNetView ArticleMATHGoogle Scholar - Rath D:
**Spaces of****-convex sequences and matrix transformations.***Indian Journal of Mathematics*1999,**41**(2):265–280.MathSciNetMATHGoogle Scholar - Rath D, Tripathy BC:
**Characterization of certain matrix operators.***Journal of Orissa Mathematical Society*1989,**8:**121–134.Google Scholar - Tripathy BC, Sen M:
**On a new class of sequences related to the space**.*Tamkang Journal of Mathematics*2002,**33**(2):167–171.MathSciNetMATHGoogle Scholar - Tripathy BC, Mahanta S:
**On a class of sequences related to the****space defined by Orlicz functions.***Soochow Journal of Mathematics*2003,**29**(4):379–391.MathSciNetMATHGoogle Scholar - Lindenstrauss J, Tzafriri L:
**On Orlicz sequence spaces.***Israel Journal of Mathematics*1971,**10:**379–390. 10.1007/BF02771656MathSciNetView ArticleMATHGoogle Scholar - Krasnoselskii MA, Rutitsky YB:
*Convex Function and Orlicz Spaces*. P.Noordhoff, Groningen, The Netherlands; 1961.Google Scholar - Bhardwaj VK, Singh N:
**Some sequence spaces defined by Orlicz functions.***Demonstratio Mathematica*2000,**33**(3):571–582.MathSciNetMATHGoogle Scholar - Güngör M, Et M, Altin Y:
**Strongly****-summable sequences defined by Orlicz functions.***Applied Mathematics and Computation*2004,**157**(2):561–571. 10.1016/j.amc.2003.08.051MathSciNetView ArticleMATHGoogle Scholar - Esi A:
**On a class of new type difference sequence spaces related to the space**.*Far East Journal of Mathematical Sciences*2004,**13**(2):167–172.MathSciNetMATHGoogle Scholar - Esi A, Et M:
**Some new sequence spaces defined by a sequence of Orlicz functions.***Indian Journal of Pure and Applied Mathematics*2000,**31**(8):967–973.MathSciNetMATHGoogle Scholar - Parashar SD, Choudhary B:
**Sequence spaces defined by Orlicz functions.***Indian Journal of Pure and Applied Mathematics*1994,**25**(4):419–428.MathSciNetMATHGoogle Scholar - Altun Y, Bilgin T:
**On a new class of sequences related to the****space defined by Orlicz function.***Taiwanese Journal of Mathematics*2009,**13**(4):1189–1196.MathSciNetMATHGoogle Scholar - Fast H:
**Sur la convergence statistique.***Colloquium Mathematicum*1951,**2:**241–244.MathSciNetMATHGoogle Scholar - Connor JS:
**The statistical and strong****-Cesàro convergence of sequences.***Analysis*1988,**8**(1–2):47–63.MathSciNetView ArticleMATHGoogle Scholar - Freedman AR, Sember JJ:
**Densities and summability.***Pacific Journal of Mathematics*1981,**95**(2):293–305.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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