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

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 .

In paper [1], the notion of -convergent and bounded sequences is introduced as follows: let be a strictly increasing sequence of positive reals tending to infinity, that is

(1.1)

We say that a sequence is -convergent to the number , called as the -limit of , if as , where

(1.2)

In particular, we say that is a -null sequence if as . Further, we say that is -bounded if . Here and in the sequel, we will use the convention that any term with a negative subscript is equal to naught, for example, and . Now, it is well known [1] that if in the ordinary sense of convergence, then

(1.3)

This implies that

(1.4)

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 introduced and studied by Sargent [2] is defined as follows:

(2.1)

Sargent [2] studied some of its properties and obtained its relationship with the space . Later on it was investigated from sequence space point of view by Rath [3], Rath and Tripathy [4], Tripathy and Sen [5], Tripathy and Mahanta [6], and others. Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define the following sequence spaces:

(2.2)

which is called an Orlicz sequence space. The space is a Banach space with the norm

(2.3)

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.

The following inequality will be used throughout the paper,

(2.4)

where and are complex numbers, and , . Tripathy and Mahanta [6] defined and studied the following sequence space. Let be an Orlicz function, then

(2.5)

Recently, Altun and Bilgin [14] defined and studied the following sequence spaces:

(2.6)

where and converges for each . In this paper, we will define the following sequence spaces:

(2.7)

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.

The space is a linear topological space paranormed by

(3.1)

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

Theorem 3.3.

if and only if

(3.2)

Proof.

Let and . Then we get

(3.3)

hence . Conversely, let us suppose that and . Then there exists a such that

(3.4)

for every . Suppose that , then there exists a sequence of natural numbers such that . Hence we can write

(3.5)

Therefore, , which is contradiction.

Corollary 3.4.

Let be an Orlicz function. Then if and only if

(3.6)

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.

Let , then there exists such that

(3.7)

for every . Let be a sequence of scalars with for all . Then from properties of Orlicz functions and seminorm, we get

(3.8)

which proves that is solid space and monotone.

4. Statistical Convergence

In [15], Fast introduced the idea of statistical convergence. This ideas was later studied by Connor [16], Freedman and Sember [17], and many others. A sequence of positive integers is called lacunary if , and as . A sequence is said to be statistically convergent to if for any ,

(4.1)

for some , where denotes the cardinality of . A sequence is said to be statistically convergent to if for any ,

(4.2)

for some .

Theorem 4.1.

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

Proof.

Let . Then

(4.3)

for every . Let be a sequence of positive numbers. Then it follows that is lacunary sequence. Then we get the following relation:

(4.4)

Taking the limit as , it follows that .

Theorem 4.2.

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

Proof.

Inclusion , is valid (from Theorem 4.1). In what follows, we will show converse inclusion. Let , since is bounded, there exists a constant such that . Then for every given , we have

(4.5)

Let us denote by , as we know this sequence is lacunary and finally we get the following relation:

(4.6)

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

5. Cesaro Convergence

In this paragraph, we will consider that is a nondecreasing sequence of positive real numbers such that , , as . Let us denote by

(5.1)

Theorem 5.1.

If is an Orlicz function. Then .

Proof.

From the definition of the sequences , it follows that . Then there exist a , such that

(5.2)

Then we get the following relation:

(5.3)

where . Knowing that and are continuous, letting on last relation, we obtain

(5.4)

Hence .

Theorem 5.2.

Let . Then for any Orlicz function, ,.

Proof.

Suppose that , then there exists such that for all . Let and , there exist such that for every

(5.5)

We can also find a constant such that

(5.6)

for all . Let be any integer with , for every . Then

(5.7)

where are sets of integer numbers which have more than elements for . Passing by limit on last relation, where (since , and ), we get that

(5.8)

from this, it follows that .

Theorem 5.3.

Let . Then for any Orlicz function, ,.