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

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.

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)

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.

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 .

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, , .