On generalized ideal asymptotically statistical equivalent of order $α$ for functions
We introduce new definitions related to the notions of asymptotically $\mathcal{I}_{\lambda}$ -statistical equivalent of order \alpha to multiple L and strongly $\mathcal{I}_{\lambda}$ -asymptotically equivalent of order $\alpha$ to multiple $L$ by using two nonnegative real-valued Lebesque measur...
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2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507491976085504 |
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| author | Öztürk, M. Savaş, R. Озтюрк, М. Саваш, Р. |
| author_facet | Öztürk, M. Savaş, R. Озтюрк, М. Саваш, Р. |
| author_sort | Öztürk, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T09:22:46Z |
| description | We introduce new definitions related to the notions of asymptotically $\mathcal{I}_{\lambda}$ -statistical equivalent of order \alpha to multiple L and
strongly $\mathcal{I}_{\lambda}$ -asymptotically equivalent of order $\alpha$ to multiple $L$ by using two nonnegative real-valued Lebesque measurable
functions in the interval $(1,\infty )$ instead of sequences. In addition, we also present some inclusion theorems. |
| first_indexed | 2026-03-24T02:10:10Z |
| format | Article |
| fulltext |
UDC 519.21
R. Savaş, M. Öztürk (Sakarya Univ., Turkey)
ON GENERALIZED IDEAL ASYMPTOTICALLY STATISTICAL EQUIVALENT
OF ORDER \bfitalpha FOR FUNCTIONS
ПРО УЗАГАЛЬНЕНИЙ IДЕАЛЬНИЙ АСИМПТОТИЧНО СТАТИСТИЧНИЙ
ЕКВIВАЛЕНТ ПОРЯДКУ \bfitalpha ДЛЯ ФУНКЦIЙ
We introduce new definitions related to the notions of asymptotically \scrI \lambda -statistical equivalent of order \alpha to multiple L and
strongly \scrI \lambda -asymptotically equivalent of order \alpha to multiple L by using two nonnegative real-valued Lebesque measurable
functions in the interval (1,\infty ) instead of sequences. In addition, we also present some inclusion theorems.
Введено новi означення, пов’язанi з поняттями асимптотично \scrI \lambda -статистичного еквiвалента порядку \alpha для кратних
L та сильно \scrI \lambda -асимптотичного еквiвалента порядку \alpha для кратних L за допомогою двох невiд’ємних дiйснознач-
них функцiй, вимiрних за Лебегом на iнтервалi (1,\infty ), замiсть послiдовностей. Крiм того, наведено також деякi
теореми про включення.
1. Introduction. This paper introduces a class of summability method that can be applied to mea-
surable functions defined on (1,\infty ). These methods are modeled on the methods of asymptotically
statistical equivalent. As part of this paper, we establish some analogs of known results for sequential
summability to the setting of real valued functions defined on (1,\infty ).
In 1993, Marouf [14] presented definitions for asymptotically equivalent sequences and asymp-
totic regular matrices. In 1997, Li [13] also presented and studied asymptotic equivalence of se-
quences and summability. In 2003, Patterson [17] extended these concepts by presenting an asymp-
totically statistical equivalent analog of these definitions and natural regularity conditions for nonne-
gative summability matrices. Recently, Savaş and Basarir [19] defined (\sigma , \lambda )-asymptotically sta-
tistical equivalent sequences. Six years later the notion of asymptotically \scrI \lambda - statistical equivalent
sequences was studied by Gümüs and Savaş [10] (see also Kumar and Sharma [12]).
A sequence (xk) is statistically convergent if ”almost all” of k its values have a common limit
point. Over the years and under different names, statistical convergence has been discussed in
number theory, trigonometric series and summability theory. Statistical convergence for sequences
was defined by Fast [8] in 1951 who provided an alternate proof of a result of Steinhaus [29] and
then reintroduced by Schoenberg [27] independently. In the latter years it was further investigated
from the sequence space point of view and linked with summability theory by Fridy [9], Connor and
Savaş [4], S̆alát [18], Cakalli [1] and many others.
The notion of statistical convergence depends on the density of subsets of \BbbN . A subset E of \BbbN
is said to have density \delta (E) if
\delta (E) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
n\sum
k=1
\chi E (k) exists.
Note that if K \subset \BbbN is a finite set, then \delta (K) = 0, and for any set K \subset \BbbN , \delta (Kc) = 1 - \delta (K).
We first recall the following definition.
c\bigcirc R. SAVAŞ, M. ÖZTÜRK, 2018
1650 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON GENERALIZED IDEAL ASYMPTOTICALLY STATISTICAL EQUIVALENT. . . 1651
Definition 1. A sequence x = (xk) is said to be statistically convergent to L if for every \varepsilon > 0
\delta (\{ k \in \BbbN : | xk - L| \geq \varepsilon \} ) = 0.
In this case we write st - \mathrm{l}\mathrm{i}\mathrm{m}x = L or xk \rightarrow L(st).
The concept of \scrI -convergence was introduced by Kostyrko et al. in a metric space [11]. Later
it was further studied by Dems [7], Das, Savaş and Ghosal [5], Gümüs and Savaş [26] and Savaş
[20 – 25] and many others. \scrI -convergence is a generalization form of statistical convergence and that
is based on the notion of an ideal of the subset of positive integers \BbbN .
On the other hand, in [2, 6] a different direction was given to the study of statistical convergence
where the notion of statistical convergence of order \alpha , 0 < \alpha < 1 was introduced by replacing n by
n\alpha in the denominator in the definition of statistical convergence. One can also see [3] for related
works.
In this paper we introduce new definitions to the notions of asymptotically
\scrI \lambda -statistical equivalent of order \alpha to multiple L and strongly \scrI \lambda -asymptotically equivalent of order
\alpha to multiple L by using two nonnegative real-valued Lebesque measurable functions x (t) and y (t)
in the interval (1,\infty ) instead of sequences. In addition, we also present some inclusion theorems.
Let \lambda = (\lambda n) be a nondecreasing sequence of positive numbers tending to \infty such that
\lambda n+1 \leq \lambda n + 1, \lambda 1 = 1.
The collection of such sequences \lambda will be denoted by \Delta .
\lambda -Statistical convergence was defined by Mursaleen [15]. In his examination he presented a
series of critical results, beginning with the following definition.
Definition 2. A sequence x = (xk) is said to be \lambda -statistically convergent or S\lambda -convergent to
the number L if for every \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda n
| \{ k \in In : | xk - L| \geq \varepsilon \} | = 0,
where In = [n - \lambda n + 1, n] for n = 1, 2, 3, . . . . and the vertical bars indicate the number of the
elements in the enclosed sets. In this case we write S\lambda - \mathrm{l}\mathrm{i}\mathrm{m}x = L or xn \rightarrow L(S\lambda ) and S\lambda denotes
the set of all \lambda -statistically convergent sequences.
Quite recently Srivastava et al. [28], studied the SL
\lambda (\scrI )-asymptotically statistical equivalent
functions.
2. Main definitions. Before we present the new definitions we shall state a few known defini-
tions.
Definition 3 [11]. A family \scrI \subset 2\BbbN is said to be an ideal of \BbbN , where \BbbN will denote the set of
all positive integers, if the following conditions hold:
(a) A, B \in \scrI implies A \cup B \in \scrI ,
(b) A \in \scrI , B \subset A implies B \in \scrI .
Definition 4. A nonempty family F \subset 2\BbbN is said to be a filter of \BbbN if the following conditions
hold:
(a) \phi /\in F,
(b) A, B \in F implies A \cap B \in F,
(c) A \in F, A \subset B implies B \in F.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1652 R. SAVAŞ, M. ÖZTÜRK
If \scrI is a proper ideal of \BbbN ( i.e., \BbbN /\in \scrI ), then the family of sets F (\scrI ) = \{ M \subset \BbbN : \exists A \in \scrI :
M = \BbbN \setminus A\} is a filter of \BbbN . It is called the filter associated with the ideal.
Definition 5. A proper ideal \scrI is said to be admissible if \{ n\} \in \scrI for each n \in \BbbN .
Throughout \scrI will stand for a proper admissible ideal of \BbbN .
Definition 6 [11]. A sequence (xn) of elements of \BbbR is said to be \scrI -convergent to L \in \BbbR if
for each \varepsilon > 0 the set A(\varepsilon ) = \{ n \in \BbbN : | xn - L| \geq \varepsilon \} \in \scrI .
Let us begin this analysis with the following preliminaries.
Definition 7 [14]. Two nonnegative sequences x = (xk) and y = (yk) are said to be asymp-
totically equivalent if
\mathrm{l}\mathrm{i}\mathrm{m}
k
xk
yk
= 1
(denoted by x \sim y).
Definition 8 [9]. The sequence x = (xk) is said to be statistically convergent to the number L
if for every \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
| \{ k \leq n : | xk - L| \geq \varepsilon \} | = 0.
In this case we write st - \mathrm{l}\mathrm{i}\mathrm{m}xk = L.
R. F. Patterson presented the following definition which is natural combination of Definitions 7
and 8.
Definition 9 [17]. Two nonnegative sequences x = (xk) and y = (yk) are said to be asymp-
totically statistical equivalent of multiple L provided that for every \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ k \leq n :
\bigm| \bigm| \bigm| \bigm| xkyk - L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = 0
(denoted by x
SL\sim y), and simply asymptotically statistical equivalent if L = 1.
The generalized La Vallée Poussin mean is defined by
tn(x) =
1
\lambda n
\sum
k\in In
xk.
A sequence x = (xk) is said to be [V, \lambda ]-summable to a number L if tn(x) \mapsto \rightarrow L and n \mapsto \rightarrow \infty .
We write
[V, \lambda ] =
\left\{ \mathrm{l}\mathrm{i}\mathrm{m}
n
1
\lambda n
\sum
k\in In
| xk - L| = 0 for some L
\right\}
for the set of sequences that are strongly summable by the La Vallée Poussin method. In the special
case where \lambda n = n, for n = 1, 2, 3, . . . , the set [V, \lambda ] reduces to the set [C, 1]-summability defined
as follows:
[C, 1] =
\Biggl\{
\mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
n\sum
k=1
| xk - L| = 0 for some L
\Biggr\}
.
We now introduce the following definitions.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON GENERALIZED IDEAL ASYMPTOTICALLY STATISTICAL EQUIVALENT. . . 1653
Definition 10. Let \lambda \in \Delta and x (t) be a nonnegative real-valued function which is measurable
in the interval (1,\infty ) . The function x (t) is said to be [V, \lambda ] (\scrI )-summable to L if
\scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda n
n\int
n - \lambda n+1
| x (t) - L| dt = 0.
If \scrI = \scrI fin = \{ A \subseteq \BbbN : A is a finite subset\} , [V, \lambda ] (\scrI )-summability becomes [V, \lambda ]-summability,
which is defined as follows (see [16]):
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda n
n\int
n - \lambda n+1
| x(t) - L| dt = 0.
Definition 11. A nonnegative real-valued function x (t) is said to be \scrI \lambda -statistically convergent
or S\lambda (\scrI ) convergent to L, if for every \varepsilon > 0 and \delta > 0,\biggl\{
n \in \BbbN :
1
\lambda n
| \{ t \in In : | x(t) - L| \geq \varepsilon \} | \geq \delta
\biggr\}
\in \scrI .
In this case we write S\lambda (\scrI ) - \mathrm{l}\mathrm{i}\mathrm{m}x(t) = L or x(t) \rightarrow L (S\lambda (\scrI )). For \scrI = \scrI fin, S\lambda (\scrI )-convergence
again coincides with \lambda -statistical convergence [16].
Following the above definitions we introduce the following new definitions related to the notions
asymptotically \scrI \lambda -statistical equivalent of multiple L, and strongly \scrI \lambda -asymptotically equivalent of
multiple L for nonnegative real-valued functions x (t) and y (t) .
3. Main results. In this section we give the main definitions and theorems of this paper.
Definition 12. Let \lambda \in \Delta and \scrI is an admissible ideal in \BbbN and x (t) , y (t) be two nonnegative
real-valued Lebesque measurable functions in the interval (1,\infty ). We say that the functions x(t) and
y(t) are strongly \scrI \lambda -asymptotically equivalent of order \alpha to L, where 0 < \alpha \leq 1, if
\scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda \alpha
n
n\int
n - \lambda n+1
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt = 0
(denoted by x(t)
V L
\lambda (\scrI )\alpha
\sim y(t)), and simply asymptotically statistical equivalent equivalent of order \alpha
if L = 1. Furthermore, let V L
\lambda (\scrI )\alpha denote the set of x(t) and y(t) such that x(t)
V L
\lambda (\scrI )\alpha
\sim y(t).
Remark 1. If \scrI = \scrI fin = \{ A \subseteq \BbbN : A is a finite subset \} , strongly \scrI \lambda -asymptotically equivalent
of order \alpha becomes strongly \lambda -asymptotically equivalent of order \alpha which is defined as follows:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\lambda \alpha
n
n\int
n - \lambda n+1
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt = 0.
Finally, for \scrI = \scrI fin and \alpha = 1 it becomes strongly \lambda -asymptotically equivalent of function [16].
We now have the following definitions.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1654 R. SAVAŞ, M. ÖZTÜRK
Definition 13. Let x (t) and y (t) be two nonnegative real-valued Lebesque measurable func-
tions in the interval (1,\infty ) and \scrI be an admissible ideal in \BbbN . We say that the functions x (t) and
y (t) are \scrI -asymptotically statistical equivalent of order \alpha to multiple L, where 0 < \alpha \leq 1, if for
every \varepsilon > 0 and \delta > 0\biggl\{
n \in \BbbN :
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ k \leq n :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\in \scrI .
In this case we write x(t)
SL(\scrI )\alpha \thicksim y(t).
Definition 14. Let \lambda \in \Delta and \scrI is an admissible ideal in \BbbN and x (t) , y (t) be two nonnegative
real-valued Lebesque measurable functions in the interval (1,\infty ) . We say that the functions x (t)
and y (t) are \scrI \lambda -asymptotically statistical equivalent of order \alpha to multiple L, where 0 < \alpha \leq 1, if
for every \varepsilon > 0 and \delta > 0\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\in \scrI
(denoted by x(t)
SL
\lambda (\scrI )\alpha
\sim y(t)), and simply asymptotically statistical equivalent of order \alpha if L = 1.
We shall denote by S\lambda (\scrI )\alpha the collection of all \scrI \lambda -asymptotically statistical equivalent of order
\alpha to multiple L.
For \scrI = \scrI fin, \scrI \lambda -asymptotically statistical equivalent of order \alpha again coincides with
\lambda -asymptotically statistical equivalent of order \alpha which is defined as follows:
Definition 15. Let \lambda \in \Delta and x(t), y(t) be two nonnegative real-valued Lebesque measurable
functions in the interval (1,\infty ) . We say that the functions x (t) and y (t) are \lambda -asymptotically
statistical equivalent of order \alpha to multiple L, where 0 < \alpha \leq 1, if for every \varepsilon > 0\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = 0
\biggr\}
.
4. Main theorems.
Theorem 1. Let 0 < \alpha \leq \beta \leq 1. Then S\lambda (\scrI )\alpha \subset S\lambda (\scrI )\beta .
Proof. Let 0 < \alpha \leq \beta \leq 1. Then for every \varepsilon > 0 we have\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
\lambda \beta
n
\leq
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
\lambda \alpha
n
and so, for any \delta > 0,\left\{ n \in \BbbN :
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
\lambda \beta
n
\geq \delta
\right\} \subset
\left\{ n \in \BbbN :
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
\lambda \alpha
n
\geq \delta
\right\} .
Hence, if the set on the right-hand side belongs to the ideal \scrI , then obviously the set on the left-hand
side also belongs to \scrI . This shows that S\lambda (\scrI )\alpha \subset S\lambda (\scrI )\beta .
Theorem 1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON GENERALIZED IDEAL ASYMPTOTICALLY STATISTICAL EQUIVALENT. . . 1655
Corollary 1. If two the functions x (t) and y (t) are \scrI \lambda -asymptotically statistical equivalent of
order \alpha to multiple L for some 0 < \alpha \leq 1, then they are \scrI \lambda -statistically convergent functions to L,
i.e., S\lambda (\scrI )\alpha \subset S\lambda (\scrI ).
Similarly we can show that the following theorems are valid.
Theorem 2. Let 0 < \alpha \leq \beta \leq 1. Then
(i) S(\scrI )\alpha \subset S(\scrI )\beta .
(ii) [V, \lambda ]\alpha (\scrI ) \subset [V, \lambda ]\beta (\scrI ) .
(iii) In particular S(\scrI )\alpha \subset S(\scrI ) and [V, \lambda ]\alpha (\scrI ) \subset [V, \lambda ] (\scrI ) .
Theorem 3. Let \lambda = \{ \lambda n\} n\in \BbbN \in \Delta . Then
(a) If x(t)
V L
\lambda (\scrI )\alpha
\sim y(t), then x(t)
SL
\lambda (\scrI )\alpha
\sim y(t).
(b) V L
\lambda (\scrI )\alpha is a proper subset of SL
\lambda (\scrI )\alpha for every ideal \scrI .
Proof. (a) Let \varepsilon > 0 and x(t)
V L
\lambda (\scrI )\alpha
\sim y(t). We have\int
t\in In
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt \geq \int
t\in In\&
\bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| >\varepsilon
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt \geq \varepsilon
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| .
So, for given \delta > 0,
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta \Rightarrow 1
\lambda \alpha
n
\int
t\in In
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt \geq \varepsilon \delta ,
i.e., \biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\subset
\subset
\left\{ n \in \BbbN :
1
\lambda \alpha
n
\left\{
\int
t\in In
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| dt \geq \varepsilon
\right\} \geq \varepsilon \delta
\right\} .
Since right-hand belongs to \scrI , then left-hand also belongs to \scrI and this completes the proof.
(b) To show that x(t)
SL
\lambda (\scrI )\alpha
\sim y(t) \varsubsetneq x(t)
V L
\lambda (\scrI )\alpha
\sim y(t), take a fixed A \in \scrI . Define a function x by
x(t) =
\left\{
t for n - [
\sqrt{}
\lambda \alpha
n] + 1 \leq t \leq n, n /\in A,
t for n - \lambda \alpha
n + 1 \leq t \leq n, n \in A,
\theta otherwise,
and y(t) = 1. Then, for every \varepsilon > 0 (0 < \varepsilon < 1),
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| =
\bigl[ \sqrt{}
\lambda \alpha
n
\bigr]
\lambda \alpha
n
\rightarrow 0
as n \rightarrow \infty and n /\in A, so, for every \delta > 0,\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\subset A \cup \{ 1, 2, . . . ,m\}
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1656 R. SAVAŞ, M. ÖZTÜRK
for some m \in N. Since \scrI is admissible so it follows that it is \scrI \lambda -asymptotically statistical equivalent
of order \alpha to multiple L. Obviously
1
\lambda \alpha
n
\int
t\epsilon In
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \rightarrow \infty (n \rightarrow \infty ) ,
i.e., it is not strongly \scrI \lambda -asymptotically equivalent of order \alpha to multiple L. Note that if A \in \scrI is
infinite, then it is not \scrI \lambda -asymptotically statistical equivalent of order \alpha of multiple L.
Theorem 4. x(t)
SL(\scrI )\alpha \thicksim y(t) implies x(t)
SL
\lambda (\scrI )\alpha
\thicksim y(t) if \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\lambda \alpha
n
n\alpha
> 0.
Proof. For given \varepsilon > 0,
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \leq n :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq 1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq
\geq \lambda \alpha
n
n\alpha
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| .
If \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\lambda \alpha
n
n\alpha
= a, then
\biggl\{
n \in \BbbN :
\lambda \alpha
n
n\alpha
<
a
2
\biggr\}
is finite. For \delta > 0,
\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\subset
\subset
\biggl\{
n \in \BbbN :
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq a
2
\delta
\biggr\}
\cup
\cup
\biggl\{
n \in \BbbN :
\lambda n
n
<
a
2
\biggr\}
.
Since \scrI is admissible, the set on the right-hand side belongs to \scrI and this completed the proof.
Theorem 5. If \lambda \in \Delta be such that \mathrm{l}\mathrm{i}\mathrm{m}n
\lambda \alpha
n
n\alpha
= 1, then x(t)
SL
\lambda (\scrI )\alpha
\thicksim y(t) implies x(t)
SL(\scrI )\alpha \thicksim y(t).
Proof. Let \delta > 0 be given. Since \mathrm{l}\mathrm{i}\mathrm{m}n
\lambda \alpha
n
n\alpha
= 1, we can choose m \in N such that
\bigm| \bigm| \bigm| \bigm| \lambda \alpha
n
n\alpha
- 1
\bigm| \bigm| \bigm| \bigm| < \delta
2
for all n \geq m. Now observe that, for \varepsilon > 0,
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \leq n :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = 1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \leq n - \lambda n :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| +
+
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq n - \lambda n
n\alpha
+
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq 1 -
\biggl(
1 - \delta
2
\biggr)
+
1
n\alpha
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| =
=
\delta
2
+
1
n\alpha
| \{ t \in In : | x(t) - L| \geq \varepsilon \} |
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON GENERALIZED IDEAL ASYMPTOTICALLY STATISTICAL EQUIVALENT. . . 1657
for all n \geq m. Hence, for any \delta > 0 we write\biggl\{
n \in \BbbN :
1
n\alpha
| \{ t \leq n : | x(t) - L| \geq \varepsilon \} | \geq \delta
\biggr\}
\subset
\subset
\biggl\{
n \in \BbbN :
1
n\alpha
| \{ t \in In : | x(t) - L| \geq \varepsilon \} | \geq \delta
2
\biggr\}
\cup \{ 1, 2, 3, . . . ,m\} .
This shows that x(t)
SL(\scrI )\alpha \thicksim y(t).
Theorem 5 is proved.
Remark 2. We do not know whether the condition in Theorem 5 is necessary and leave it is an
open problem.
Now we shall prove a more inclusion relation theorem.
Theorem 6. Let \lambda = (\lambda n) and \mu = (\mu n) be two functions in \Delta such that \lambda n \leq \mu n for all
n \in \BbbN and let \alpha and \beta be fixed real numbers such that 0 < \alpha \leq \beta \leq 1.
(i) If
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\lambda \alpha
n
\mu \beta
n
> 0, (1)
then SL
\mu (\scrI )\beta \subseteq SL
\lambda (\scrI )\alpha .
(ii) If
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mu n
\lambda \beta
n
= 1, (2)
then SL
\lambda (\scrI )\alpha \subseteq SL
\mu (\scrI )\beta .
Proof. (i) Suppose that \lambda n \leq \mu n for all n \in \BbbN and let (1) be satisfied. For given \varepsilon > 0 we
have \biggl\{
t \in Jn :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\}
\supseteq
\biggl\{
t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\}
,
where In = [n - \lambda n + 1, n] and Jn = [n - \mu n + 1, n]. Therefore we can write
1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in Jn :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \lambda \alpha
n
\mu \beta
n
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
and so, for all n \in \BbbN and for any \delta > 0, we obtain\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\subseteq
\subseteq
\biggl\{
n \in \BbbN :
1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in Jn :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\lambda \alpha
n
\mu \beta
n
\biggr\}
\in \scrI .
Hence, SL
\mu (\scrI )\beta \subseteq SL
\lambda (\scrI )\alpha .
(ii) Let x = (xk) and y = (yk) \in SL
\lambda (\scrI )\alpha and (2) be satisfied. Since In \subset Jn, for \varepsilon > 0 we
may write
1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in Jn :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = 1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ n - \mu n < t \leq n - \lambda n :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| +
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1658 R. SAVAŞ, M. ÖZTÜRK
+
1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq \mu n - \lambda n
\mu \beta
n
+
1
\lambda \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq
\Biggl(
\mu n - \lambda \beta
n
\lambda \beta
n
\Biggr)
+
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl(
\mu n
\lambda \beta
n
- 1
\biggr)
+
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm|
for all n \in \BbbN . Hence, for any \delta > 0, we have\biggl\{
n \in \BbbN :
1
\mu \beta
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in Jn :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\subseteq
\subseteq
\biggl\{
n \in \BbbN :
1
\lambda \alpha
n
\bigm| \bigm| \bigm| \bigm| \biggl\{ t \in In :
\bigm| \bigm| \bigm| \bigm| x(t)y(t)
- L
\bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| \geq \delta
\biggr\}
\in \scrI .
This implies that SL
\lambda (\scrI )\alpha \subseteq SL
\mu (\scrI )\beta .
Theorem 6 is proved.
From Theorem 6 we have the following corollaries.
Corollary 2. Let \lambda = (\lambda n) and \mu = (\mu n) be two sequences in \Delta such that \lambda n \leq \mu n for all
n \in \BbbN . If (1) holds, then
(i) SL
\mu (\scrI )\alpha \subseteq SL
\lambda (\scrI )\alpha for each \alpha \in (0, 1] ,
(ii) SL
\mu (\scrI ) \subseteq SL
\lambda (\scrI )\alpha for each \alpha \in (0, 1] ,
(iii) SL
\mu (\scrI ) \subseteq SL
\lambda (\scrI ).
Corollary 3. Let \lambda = (\lambda n) and \mu = (\mu n) be two sequences in \Delta such that \lambda n \leq \mu n for all
n \in \BbbN . If (2) holds, then
(i) SL
\lambda (\scrI )\alpha \subseteq SL
\mu (\scrI )\alpha for each \alpha \in (0, 1] ,
(ii) SL
\lambda (\scrI )\alpha \subseteq SL
\mu (\scrI ) for each \alpha \in (0, 1] ,
(iii) SL
\lambda (\scrI ) \subseteq SL
\mu (\scrI ).
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Received 27.12.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
|
| id | umjimathkievua-article-1666 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:10Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/54/b91411f56900a7e15e74a7da1267a754.pdf |
| spelling | umjimathkievua-article-16662019-12-05T09:22:46Z On generalized ideal asymptotically statistical equivalent of order $α$ for functions Про узагальнений iдеальний асимптотично статистичний еквiвалент порядку $\alpha$ для функцiй Öztürk, M. Savaş, R. Озтюрк, М. Саваш, Р. We introduce new definitions related to the notions of asymptotically $\mathcal{I}_{\lambda}$ -statistical equivalent of order \alpha to multiple L and strongly $\mathcal{I}_{\lambda}$ -asymptotically equivalent of order $\alpha$ to multiple $L$ by using two nonnegative real-valued Lebesque measurable functions in the interval $(1,\infty )$ instead of sequences. In addition, we also present some inclusion theorems. Введено новi означення, пов’язанi з поняттями асимптотично $\mathcal{I}_{\lambda}$ -статистичного еквiвалента порядку $\alpha$ для кратних $L$ та сильно $\mathcal{I}_{\lambda}$ -асимптотичного еквiвалента порядку $\alpha$ для кратних $L$ за допомогою двох невiд’ємних дiйснозначних функцiй, вимiрних за Лебегом на iнтервалi $(1,\infty )$, замiсть послiдовностей. Крiм того, наведено також деякi теореми про включення. Institute of Mathematics, NAS of Ukraine 2018-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1666 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 12 (2018); 1650-1659 Український математичний журнал; Том 70 № 12 (2018); 1650-1659 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1666/648 Copyright (c) 2018 Öztürk M.; Savaş R. |
| spellingShingle | Öztürk, M. Savaş, R. Озтюрк, М. Саваш, Р. On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title | On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title_alt | Про узагальнений iдеальний асимптотично статистичний
еквiвалент порядку $\alpha$ для функцiй |
| title_full | On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title_fullStr | On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title_full_unstemmed | On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title_short | On generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| title_sort | on generalized ideal asymptotically statistical equivalent of order $α$ for functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1666 |
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