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A New Class of Sequences Related to the
Spaces Defined by Sequences of Orlicz Functions
Journal of Inequalities and Applications volume 2011, Article number: 539745 (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
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ1_HTML.gif)
We say that a sequence is
-convergent to the number
, called as the
-limit of
, if
as
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ3_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ5_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ6_HTML.gif)
which is called an Orlicz sequence space. The space is a Banach space with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ7_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ8_HTML.gif)
where and
are complex numbers, and
,
. Tripathy and Mahanta [6] defined and studied the following sequence space. Let
be an Orlicz function, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ9_HTML.gif)
Recently, Altun and Bilgin [14] defined and studied the following sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ10_HTML.gif)
where and converges for each
. In this paper, we will define the following sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ12_HTML.gif)
In what follows, we will show inclusion theorems between spaces .
Theorem 3.3.
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ13_HTML.gif)
Proof.
Let and
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ14_HTML.gif)
hence . Conversely, let us suppose that
and
. Then there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ15_HTML.gif)
for every . Suppose that
, then there exists a sequence of natural numbers
such that
. Hence we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ16_HTML.gif)
Therefore, , which is contradiction.
Corollary 3.4.
Let be an Orlicz function. Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ18_HTML.gif)
for every . Let
be a sequence of scalars with
for all
. Then from properties of Orlicz functions and seminorm, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ19_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ20_HTML.gif)
for some , where
denotes the cardinality of
. A sequence
is said to be
statistically convergent to
if for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ21_HTML.gif)
for some .
Theorem 4.1.
If is any Orlicz function,
strictly increasing sequence, then
.
Proof.
Let . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ22_HTML.gif)
for every . Let
be a sequence of positive numbers. Then it follows that
is lacunary sequence. Then we get the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ24_HTML.gif)
Let us denote by , as we know this sequence is lacunary and finally we get the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ27_HTML.gif)
Then we get the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ28_HTML.gif)
where . Knowing that
and
are continuous, letting
on last relation, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ30_HTML.gif)
We can also find a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ31_HTML.gif)
for all . Let
be any integer with
, for every
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ32_HTML.gif)
where are sets of integer numbers which have more than
elements for
. Passing by limit on last relation, where
(since
,
and
), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F539745/MediaObjects/13660_2010_Article_2348_Equ33_HTML.gif)
from this, it follows that .
Theorem 5.3.
Let . Then for any Orlicz function,
,
.
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Braha, N.L. A New Class of Sequences Related to the Spaces Defined by Sequences of Orlicz Functions.
J Inequal Appl 2011, 539745 (2011). https://doi.org/10.1155/2011/539745
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DOI: https://doi.org/10.1155/2011/539745