f-asymptotically lacunary ideal equivalence of double sequences

Abstract

In this study, we present the notions of f-asymptotically $\mathcal{I}_{2}$-equivalence, strongly f-asymptotically $\mathcal{I}_{2}$-equivalence, f-asymptotically lacunary $\mathcal{I}_{2}$-equivalence, and strongly f-asymptotically lacunary $\mathcal{I}_{2}$-equivalence of double sequences and investigate some relationships between them. Also, we examine some relationships between strongly f-asymptotically $\mathcal{I}_{2}$-equivalence and asymptotically $\mathcal{I}_{2}$-statistical equivalence and between strongly f-asymptotically lacunary $\mathcal{I}_{2}$-equivalence and asymptotically lacunary $\mathcal{I}_{2}$-statistical equivalence of double sequences.

Introduction and background

Throughout the paper, $\mathbb{N}$ is the set of all positive integers, and $\mathbb{R}$ is the set of all real numbers. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [14] and Schoenberg [39]. Fridy and Orhan [15] studied lacunary statistical convergence. This concept was extended to the double sequences by Mursaleen and Edely [28]. A lot of development have been made in this area after the works of [1, 2, 13, 25,26,27].

The idea of $\mathcal{I}$-convergence was introduced by Kostyrko, S̆alát, and Wilczyński [20] as a generalization of statistical convergence. Das et al. [4] introduced the concept of $\mathcal{I}$-convergence of double sequences in a metric space and studied some properties of this convergence. A lot of development have been made in this area after the works of [5,6,7,8,9,10,11, 29,30,31,32, 40, 44].

Marouf [24] presented definitions for asymptotically equivalent and asymptotic regular matrices. Patterson [35] presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Dündar et al. [12] defined asymptotically $\mathcal{I}^{\sigma }_{2}$-equivalent, asymptotically invariant equivalent, strongly asymptotically invariant equivalent, and p-strongly asymptotically invariant equivalent for double sequences. Ulusu and Dündar [42] introduced the concepts of asymptotically lacunary $\mathcal{I}_{2}$-invariant equivalence, asymptotically lacunary $\sigma _{2}$-equivalence, and asymptotically lacunary invariant $S_{2}$-equivalence of double sequences. Hazarika and Kumar [16] studied asymptotically double lacunary statistically equivalent sequences in ideal context.

The modulus function was introduced by Nakano [33]. Maddox [23], Pehlivan [37], and many authors used the modulus function f to define some new concepts and inclusion theorems. Kumar and Sharma [21] studied lacunary equivalent sequences by ideals and the modulus function. Also, several authors define some new concepts and give inclusion theorems using a modulus function f (see [17, 18]).

Now we recall the basic concepts and some definitions (see [3, 15, 19, 20, 22,23,24, 34,35,36,37,38, 41, 43]).

By a lacunary sequence we mean an increasing integer sequence $\theta =\{k_{r}\}$ such that $k_{0}=0$ and $h_{r}=k_{r}-k _{r-1}\rightarrow \infty$ as $r\rightarrow \infty$. Throughout the paper, we let $\theta =\{k_{r}\}$ be a lacunary sequence.

A double sequence $\theta _{2} = \{(k_{r},j_{u})\}$ is called a double lacunary sequence if there exist two increasing sequences of integers such that $k_{0}=0$, $h_{r}= k_{r}-k_{r-1}\rightarrow \infty$ and $j_{0}=0$, $\bar{h}_{u}= j_{u}-j_{u-1}\rightarrow \infty$ as $r,u\rightarrow \infty$. We further use the following notations:

$$\begin{aligned}& k_{ru}= k_{r}j_{u}, \qquad h_{ru}= h_{r}\bar{h}_{u}, \qquad I_{ru}=\bigl\{ (k,j): k_{r-1}< k\leq k_{r} \textrm{ and } j_{u-1}< j\leq j_{u}\bigr\} , \\& q_{r}=\frac{k_{r}}{k_{r-1}} \quad \textrm{and}\quad q_{u}= \frac{j_{u}}{j_{u-1}}. \end{aligned}$$

Throughout the paper, we let $\theta _{2} = \{(k_{r},j_{u})\}$ be a double lacunary sequence.

A family of sets $\mathcal{I}\subseteq 2^{\mathbb{N}}$ is called an ideal if and only if

• (i) $\emptyset \in \mathcal{I}$,

• (ii) for all $A,B\in \mathcal{I}$, we have $A\cup B\in \mathcal{I}$,

• (iii) for each $A\in \mathcal{I}$ and each $B\subseteq A$, we have $B\in \mathcal{I}$.

An ideal is called nontrivial if $\mathbb{N}\notin \mathcal{I}$, and a nontrivial ideal is called admissible if $\{ n \}\in \mathcal{I}$ for each $n\in \mathbb{N}$. Throughout the paper, we let $\mathcal{I}$ be an admissible ideal.

A nontrivial ideal $\mathcal{I}_{2}$ of $\mathbb{N} \times \mathbb{N}$ is called a strongly admissible ideal if $\{i\}\times \mathbb{N}$ and $\mathbb{N}\times \{i\}$ belong to $\mathcal{I}_{2}$ for each $i \in N$.

Throughout the paper, we let $\mathcal{I}_{2}$ be a strongly admissible ideal in $\mathbb{N} \times \mathbb{N}$.

It is evident that a strongly admissible ideal is admissible.

We denote $\mathcal{I}_{2}^{0} = \{A \subset \mathbb{N}\times \mathbb{N}: (\exists m(A)\in \mathbb{N})\ (i,j \geq m(A) \Rightarrow (i,j) \notin A)\}$. Then $\mathcal{I}_{2}^{0}$ is a strongly admissible ideal, and clearly an ideal $\mathcal{I}_{2}$ is strongly admissible if and only if $\mathcal{I}_{2}^{0} \subset \mathcal{I}_{2}$.

Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be asymptotically equivalent if $\lim_{k\rightarrow \infty }\frac{x _{k}}{y_{k}}=1$ (denoted by $x\sim y$).

Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be asymptotically statistically equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{n} \biggl\vert \biggl\{ k \leq n: \biggl\vert \frac{x_{k}}{y_{k}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert =0 \end{aligned}$$

(denoted by $x\overset{S_{L}}{\sim }y$) and simply asymptotically statistically equivalent, if $L=1$.

Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be strongly asymptotically equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for every $\varepsilon >0$,

$$\begin{aligned} \Biggl\{ n\in \mathbb{N}:\frac{1}{n}\sum_{k=1}^{n} \biggl\vert \frac{x_{k}}{y _{k}}-L \biggr\vert \geq \varepsilon \Biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I(\omega )}}}{\sim }y_{k}$) and simply strongly asymptotically equivalent with respect to the ideal $\mathcal{I}$ if $L=1$.

Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be strongly asymptotically lacunary equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ r\in \mathbb{N}:\frac{1}{h_{r}}\sum_{k\in I_{r}} \biggl\vert \frac{x _{k}}{y_{k}}-L \biggr\vert \geq \varepsilon \biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I}(N_{\theta })}}{\sim }y_{k}$) and simply strongly asymptotically lacunary $\mathcal{I}$-equivalent with respect to the ideal $\mathcal{I}$ if $L=1$.

Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be asymptotically lacunary statistically equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for all $\varepsilon >0$ and $\gamma >0$,

$$\begin{aligned} \biggl\{ r\in \mathbb{N}:\frac{1}{h_{r}} \biggl\vert \biggl\{ k\in I_{r}: \biggl\vert \frac{x_{k}}{y_{k}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \geq \gamma \biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I(S_{\theta })}}}{\sim }y_{k}$) and simply asymptotically lacunary $\mathcal{I}$-statistically equivalent if $L=1$.

A function $f:[0,\infty )\rightarrow [0,\infty )$ is called a modulus if

• 1. $f(x)=0$ if and if only if $x=0$,

• 2. $f(x+y)\leq f(x)+f(y)$,

• 3. f is increasing, and

• 4. f is continuous from the right at 0.

A modulus may be unbounded (for example, $f(x)=x^{p}$, $0< p<1$) or bounded (for example, $f(x)=\frac{x}{x+1}$).

Let f be a modulus function. Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be f-asymptotically equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ k\in \mathbb{N}:f \biggl( \biggl\vert \frac{x_{k}}{y_{k}}-L \biggr\vert \biggr) \geq \varepsilon \biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I}(f)}}{\sim }y_{k}$) and simply f-asymptotically $\mathcal{I}$-equivalent if $L=1$.

Let f be a modulus function. Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be strongly f-asymptotically equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for every $\varepsilon >0$,

$$\begin{aligned} \Biggl\{ n\in \mathbb{N}:\frac{1}{n}\sum_{k=1}^{n}f \biggl( \biggl\vert \frac{x _{k}}{y_{k}}-L \biggr\vert \biggr)\geq \varepsilon \Biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I}(\omega _{f})}}{\sim }y_{k}$)) and simply strongly f-asymptotically $\mathcal{I}$-equivalent if $L=1$.

Let f be a modulus function. Two nonnegative sequences $x=(x_{k})$ and $y=(y_{k})$ are said to be strongly f-asymptotically lacunary equivalent of multiple L with respect to the ideal $\mathcal{I}$ if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ r\in \mathbb{N}:\frac{1}{h_{r}}\sum_{k\in I_{r}}f \biggl( \biggl\vert \frac{x _{k}}{y_{k}}-L \biggr\vert \biggr)\geq \varepsilon \biggr\} \in \mathcal{I} \end{aligned}$$

(denoted by $x_{k}\overset{{\mathcal{I}(N_{\theta }^{f})}}{\sim }y _{k}$)) and simply strongly f-asymptotically lacunary $\mathcal{I}$-equivalent if $L=1$.

Two nonnegative double sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be asymptotically strongly $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}: \frac{1}{mn}\sum _{k,j=1} ^{m,n} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \Biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{[{\mathcal{I}^{L}_{2}}]}{\sim }y_{kj}$) and simply asymptotically $\mathcal{I}_{2}$-statistical equivalent if $L=1$.

Two nonnegative double sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be asymptotically $\mathcal{I}_{2}$-statistically equivalent of multiple L if for all $\varepsilon >0$ and each $\gamma >0$,

$$\begin{aligned} \biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}:\frac{1}{mn} \biggl\vert \biggl\{ k,j\leq m,n: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \geq \gamma \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}$) and simply asymptotically $\mathcal{I}_{2}$-statistically equivalent if $L=1$.

Two nonnegative double sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be asymptotically lacunary $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h_{ru}} \sum _{(k,j)\in I_{ru}} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y _{kj}$) and simply strongly asymptotically lacunary $\mathcal{I}_{2}$-equivalent if $L=1$.

Two nonnegative double sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be asymptotically lacunary $\mathcal{I}_{2}$-statistically equivalent of multiple L if for all $\varepsilon >0$ and $\gamma >0$,

$$\begin{aligned} \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}:\frac{1}{h_{ru}} \biggl\vert \biggl\{ (k,j)\in I_{ru}: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \geq \gamma \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{{\mathcal{I}_{2}(S_{\theta })}}{\sim }y _{kj}$) and simply asymptotically $\mathcal{I}_{2}$-statistically equivalent if $L=1$.

Lemma 1

([37])

Let f be a modulus, and let $0<\delta <1$. Then, for each $x\geq \delta$, we have $f(x)\leq 2f(1)\delta ^{-1}x$.

Main results

Definition 2.1

Let f be a modulus function. Two nonnegative sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be f-asymptotically $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ (k,j)\in \mathbb{N}\times \mathbb{N} :f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \varepsilon \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{{\mathcal{I}_{2}^{L}}(f)}{\sim }y_{kj}$) and simply f-asymptotically $\mathcal{I}_{2}$-equivalent if $L=1$.

Definition 2.2

Let f be a modulus function. The two nonnegative sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be strongly f-asymptotically $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}: \frac{1}{mn}\sum _{k,j=1} ^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr)\geq \varepsilon \Biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$) and simply strongly f-asymptotically $\mathcal{I}_{2}$-equivalent if $L=1$.

Theorem 2.1

Let f be a modulus function. Then $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}\Rightarrow x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$.

Proof

Suppose that $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}$, and let $\varepsilon >0$ be given. Select $0<\delta <1$ such that $f(t)<\varepsilon$ for $0\leq t\leq \delta$. We can write

$$\begin{aligned} \frac{1}{mn}\sum_{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) = & \frac{1}{mn} \sum_{\substack{k,j=1\\ \vert \frac{x _{kj}}{y_{kj}}-L \vert \leq \delta }}^{m,n} f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ & {}+ \frac{1}{mn} {\sum_{\substack{k,j=1\\\vert \frac{x_{kj}}{y_{kj}}-L \vert > \delta}}^{m,n}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr), \end{aligned}$$

and so by Lemma 1

$$\begin{aligned} \frac{1}{mn}\sum_{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) < \varepsilon + \biggl(\frac{2f(1)}{\delta } \biggr) \frac{1}{mn} \sum _{k,j=1}^{m,n} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert . \end{aligned}$$

Thus, for any $\gamma >0$,

$$\begin{aligned} & \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}:\frac{1}{mn}\sum _{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \gamma \Biggr\} \\ & \quad \subseteq \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}: \frac{1}{mn} \sum_{k,j=1}^{m,n} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \frac{(\gamma -\varepsilon )\delta }{2f(1)} \Biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}$, it follows that the second set and thus the first set in the above expression belong to $\mathcal{I}_{2}$. This proves that $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$. □

Theorem 2.2

If $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=\alpha >0$, then $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}\Leftrightarrow x_{kj}\overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}$.

Proof

We showed that $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj} \Rightarrow x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$ in Theorem 2.1. Now we must show that $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}\Rightarrow x_{kj} \overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}$.

Let $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=\alpha >0$. Then we have $f(t)\geq \alpha t$ for all $t\geq 0$. Assume that $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$. Since

$$\begin{aligned} \frac{1}{mn}\sum_{k,j=1}^{m,n} f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) \geq & \frac{1}{mn}{\sum_{k,j=1}^{m,n}} \alpha \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) =\alpha \Biggl(\frac{1}{mn}{\sum_{k,j=1}^{m,n}} \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \Biggr), \end{aligned}$$

it follows that for each $\varepsilon >0$, we have

$$\begin{aligned} & \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}:\frac{1}{mn}\sum _{k,j=1}^{m,n} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \Biggr\} \\ & \quad \subseteq \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}: \frac{1}{mn} \sum_{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \alpha \varepsilon \Biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$, it follows that the latter set and hence the former set in the above expression belong to $\mathcal{I}_{2}$. This proves that $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}\Leftrightarrow x _{kj}\overset{[{\mathcal{I}_{2}^{L}}]}{\sim }y_{kj}$. □

Definition 2.3

Let f be a modulus function. Two nonnegative sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be f-asymptotically lacunary $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ (k,j)\in I_{ru}:f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \varepsilon \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj} \overset{{\mathcal{I}^{L}_{\theta _{2}}}(f)}{\sim }y_{kj}$)) and simply f-asymptotically lacunary $\mathcal{I}_{2}$-equivalent if $L=1$.

Definition 2.4

Let f be a modulus function. Two nonnegative sequences $x=(x_{kj})$ and $y=(y_{kj})$ are said to be strongly f-asymptotically lacunary $\mathcal{I}_{2}$-equivalent of multiple L if for every $\varepsilon >0$,

$$\begin{aligned} \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h_{ru}} \sum _{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \varepsilon \biggr\} \in \mathcal{I}_{2} \end{aligned}$$

(denoted by $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$)) and simply strongly f-asymptotically lacunary $\mathcal{I}_{2}$-equivalent if $L=1$.

Theorem 2.3

Let f be a modulus function. Then, $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}\Rightarrow x _{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$.

Proof

Let $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}$, and let $\varepsilon >0$ be given. Choose $0<\delta <1$ such that $f(t)<\varepsilon$ for $0\leq t\leq \delta$. We can write

$$\begin{aligned} \frac{1}{h_{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) = &\frac{1}{h_{ru}} {\sum _{\substack{(k,j)\in I _{ru}\\\vert \frac{x _{kj}}{y_{kj}}-L \vert \leq \delta}}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ &{} + \frac{1}{h_{ru}} {\sum_{\substack{(k,j)\in I_{ru}\\\vert \frac{x_{kj}}{y_{kj}}-L \vert > \delta}}}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr), \end{aligned}$$

and so by Lemma 1

$$ \frac{1}{h_{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr)< \varepsilon + \biggl( \frac{2f(1)}{\delta } \biggr)\frac{1}{h_{ru}}\sum _{(k,j)\in I_{r}} \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert . $$

Thus, for each any $\gamma >0$,

$$\begin{aligned} & \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h_{ru}} \sum _{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \gamma \biggr\} \\ & \quad \subseteq \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h _{ru}}\sum_{(k,j)\in I_{ru}} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \frac{(\gamma -\varepsilon )\delta }{2f(1)} \biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}$, it follows that the latter set and hence the former set in the above expression belong to $\mathcal{I}_{2}$. This proves that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$. □

Theorem 2.4

If $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=\alpha >0$, then $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj} \Leftrightarrow x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}$.

Proof

We showed that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}\Rightarrow x _{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$ in Theorem 2.3. Now we must show that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}\Rightarrow x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}$.

Let $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=\alpha >0$. Then we have $f(t)\geq \alpha t$ for all $t\geq 0$. Assume that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$. From

$$\begin{aligned} \frac{1}{h_{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq & \frac{1}{h_{ru}}{\sum_{(k,j)\in I_{ru}}}\alpha \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ & \\ =& \alpha \biggl(\frac{1}{h_{ru}}{\sum_{(k,j)\in I_{ru}}} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \end{aligned}$$

it follows that for each $\varepsilon >0$, we have

$$\begin{aligned} & \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}:\frac{1}{h_{ru}} \sum _{(k,j)\in I_{ru}} \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \\ & \quad \subseteq \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h _{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \alpha \varepsilon \biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y _{kj}$, it follows that the latter set and hence the former set in the above expression belong to $\mathcal{I}_{2}$. This proves that $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}]}{\sim }y_{kj}\Leftrightarrow x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$. □

Theorem 2.5

Let f be a modulus function. If $\liminf_{r,u}q_{r,u}>1$, then

$$\begin{aligned} x_{kj}\overset{\bigl[{\mathcal{I}_{2}^{L}}(f) \bigr]}{\sim }y_{kj}\quad \Rightarrow\quad x _{kj}\overset{\bigl[{ \mathcal{I}_{\theta _{2}}^{L}}(f)\bigr]}{\sim }y_{kj}. \end{aligned}$$

Proof

Suppose that $\liminf_{r,u}q_{r,u}>1$. Then there exists $\eta >0$ such that $q_{r,u}\geq 1+\eta$ for sufficiently large r, u. Then we have

$$\begin{aligned} \frac{h_{ru}}{k_{r}j_{u}}\geq \frac{\eta }{1+\eta }. \end{aligned}$$

Let $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$. For sufficiently large r, u, we have

$$\begin{aligned} \frac{1}{k_{r}j_{u}}\sum_{k,j=1,1}^{k_{r}j_{u}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq & \frac{1}{k_{r}j_{u}}{\sum_{(k,j)\in I_{ru}}} f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ = & \biggl(\frac{h_{ru}}{k_{r}j_{u}} \biggr)\frac{1}{h_{ru}}{\sum _{(k,j)\in I_{ru}}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ \geq & \frac{\eta }{1 +\eta }\frac{1}{h_{ru}} {\sum _{(k,j)\in I_{ru}}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr), \end{aligned}$$

which gives, for any $\varepsilon >0$,

$$\begin{aligned} & \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}:\frac{1}{h_{ru}} \sum _{(k,j)\in I_{ru}} f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr)\geq \varepsilon \biggr\} \\ & \quad \subseteq \Biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{k _{r}j_{u}}\sum_{k,j=1,1}^{k_{r}j_{u}}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) \geq \frac{\varepsilon \eta }{1+\eta } \Biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$, it follows that the latter set and hence the former set belong to $\mathcal{I}_{2}$. This shows that $x_{kj} \overset{[{\mathcal{I}_{\theta _{2}}^{L}}(f)]}{\sim }y_{kj}$. □

Theorem 2.6

Let f be a modulus function. Then, $x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}\Rightarrow x_{kj} \overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}$.

Proof

Assume that $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$, and let $\varepsilon >0$ be given. From

$$\begin{aligned} \frac{1}{mn}\sum_{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) \geq & \frac{1}{mn} {\sum_{\substack{k,j=1\\\vert \frac{x _{kj}}{y_{kj}}-L \vert \geq \varepsilon}}^{n}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ \geq & f(\varepsilon )\cdot\frac{1}{mn} \biggl\vert \biggl\{ k\leq m, j \leq n: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \end{aligned}$$

it follows that for any $\gamma >0$, we have

$$\begin{aligned}& \biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}:\frac{1}{mn} \biggl\vert \biggl\{ k\leq m, j\leq n: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \geq \frac{\gamma }{f(\varepsilon )} \biggr\} \\& \quad \subseteq \Biggl\{ (m,n)\in \mathbb{N}\times \mathbb{N}:\frac{1}{mn} \sum_{k,j=1}^{m,n} f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr)\geq \gamma \Biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$, it follows the latter set and hence the former set in the above expression belong to $\mathcal{I}_{2}$. Therefore $x_{kj} \overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}$. □

Theorem 2.7

Let f be a modulus function. If f is bounded, then $x_{kj} \overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}\Leftrightarrow x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$.

Proof

We showed that $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y _{kj}\Rightarrow x_{kj}\overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}$ in Theorem 2.6. Now we must show that $x_{kj} \overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}\Rightarrow x_{kj} \overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y_{kj}$.

Assume that f is bounded and let $x_{kj} \overset{{\mathcal{I}_{2}(S)}}{\sim }y_{kj}$. Since f is bounded, there exists a positive real number M such that $|f(x)|\leq M$ for all $x\geq 0$. We have

$$\begin{aligned} \frac{1}{mn}\sum_{k,j=1}^{m,n}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) = & \frac{1}{mn} {\sum_{\substack{k,j=1\\\vert \frac{x_{kj}}{y _{kj}}-L \vert \geq \varepsilon}}^{m,n}} f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ &{} +\frac{1}{mn} {\sum_{\substack{k,j=1\\\vert \frac{x_{kj}}{y_{kj}}-L \vert < \varepsilon}}^{m,n}} f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) \\ \leq & \frac{M}{mn} \biggl\vert \biggl\{ k\leq m, j\leq n: \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert +f( \varepsilon ). \end{aligned}$$

This proves that $x_{kj}\overset{[{\mathcal{I}_{2}^{L}}(f)]}{\sim }y _{kj}$. □

Theorem 2.8

Let f be a modulus function. Then $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}\Rightarrow x_{kj}\overset{{\mathcal{I}(S_{\theta _{2})}}}{\sim }y_{kj}$.

Proof

Assume that $x_{k} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{k}$, and let $\varepsilon >0$ be given. From

$$\begin{aligned} \frac{1}{h_{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq & \frac{1}{h_{ru}} {\sum_{\substack{(k,j)\in I_{ru}\\\vert \frac{x_{kj}}{y_{kj}}-L \vert \geq \varepsilon}}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ \geq & f(\varepsilon )\cdot\frac{1}{h_{ru}} \biggl\vert \biggl\{ (k,j) \in I_{ru}: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \end{aligned}$$

it follows that for any $\gamma >0$,

$$\begin{aligned}& \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}:\frac{1}{h_{ru}} \biggl\vert \biggl\{ (k,j)\in I_{ru}: \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert \geq \gamma \biggr\} \\& \quad \subseteq \biggl\{ (r,u)\in \mathbb{N}\times \mathbb{N}: \frac{1}{h _{ru}} \sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \geq \gamma f(\varepsilon ) \biggr\} . \end{aligned}$$

Since $x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y _{kj}$, the last set belongs to $\mathcal{I}_{2}$, and so by the definition of an ideal the first set belongs to $\mathcal{I}_{2}$. Therefore $x_{kj}\overset{{\mathcal{I}(S_{\theta _{2}})}}{\sim }y_{kj}$. □

Theorem 2.9

Let f be a modulus function. If f is bounded, then $x_{kj} \overset{{\mathcal{I}(S_{\theta _{2}})}}{\sim }y_{kj}\Leftrightarrow x_{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$.

Proof

We showed that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}\Rightarrow x_{kj}\overset{{\mathcal{I}(S_{\theta _{2}})}}{\sim }y_{kj}$ in Theorem 2.8. Now we must show that $x_{kj} \overset{{\mathcal{I}(S_{\theta _{2}})}}{\sim }y_{kj}\Rightarrow x _{kj}\overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$.

Assume that f is bounded and let $x_{kj} \overset{{\mathcal{I}(S_{\theta _{2}})}}{\sim }y_{kj}$. Since f is bounded, there exists a positive real number M such that $|f(x)| \leq M$ for all $x\geq 0$. We have

$$\begin{aligned} \frac{1}{h_{ru}}\sum_{(k,j)\in I_{ru}}f \biggl( \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \biggr) = & \frac{1}{h_{ru}} {\sum_{\substack{(k,j)\in I_{ru}\\\vert \frac{x _{kj}}{y_{kj}}-L \vert \geq \varepsilon}}}f \biggl( \biggl\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert \biggr) \\ &{} +\frac{1}{h_{ru}} {\sum_{\substack{(k,j)\in I_{ru}\\\vert \frac{x_{kj}}{y_{kj}}-L \biggr\vert < \varepsilon}}}f \biggl( \biggl\vert \frac{x_{kj}}{y _{kj}}-L \biggr\vert \biggr) \\ \leq & \frac{M}{h_{ru}} \biggl\vert \biggl\{ (k,j)\in I_{ru}: \biggl\vert \frac{x _{kj}}{y_{kj}}-L \biggr\vert \geq \varepsilon \biggr\} \biggr\vert +f( \varepsilon ) . \end{aligned}$$

This proves that $x_{kj} \overset{[{\mathcal{I}^{L}_{\theta _{2}}}(f)]}{\sim }y_{kj}$. □