# $$f_{(\lambda,\mu)}$$-statistical convergence of order α̃ for double sequences

## Abstract

New concepts of $$f_{\lambda,\mu }$$-statistical convergence for double sequences of order α̃ and strong $$f_{\lambda,\mu }$$-Cesàro summability for double sequences of order α̃ are introduced for sequences of (complex or real) numbers. Furthermore, we give the relationship between the spaces $$w_{\tilde{\alpha },0}^{2} ( f,\lambda,\mu )$$, $$w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu )$$ and $$w_{\tilde{\alpha},\infty }^{2} ( f,\lambda,\mu )$$. Then we express the properties of strong $$f_{\lambda,\mu }$$-Cesàro summability of order β̃ which is related to strong $$f_{\lambda,\mu }$$-Cesàro summability of order α̃. Also, some relations between $$f_{\lambda,\mu }$$-statistical convergence of order α̃ and strong $$f_{\lambda,\mu }$$-Cesàro summability of order α̃ are given.

## Introduction

The first idea of statistical convergence goes back to the first edition of the famous Zygmund’s monograph . The statistical convergence was introduced for real and complex sequences by Steinhaus . Fast  extended the usual concept of sequential limit and called it statistical convergence. Schoenberg  called it as D-Convergence. The idea depends on a certain density of subsets of $$\mathbb{N}$$. The natural density (or asymptotic density) of a set A $$\mathbb{N}$$ is defined by $$\delta ( A ) =\lim_{n\rightarrow \infty } \frac{1}{n}\vert \{ k\leq n:k\in A \} \vert$$ if the limit exists, where $$\vert A(n)\vert$$ is cardinality of the set $$A(n)$$ (see ). A sequence x= $$( x_{k} )$$ of complex numbers is said to be statistically convergent to some number if $$\delta ( \{ k\in \mathbb{N}:\vert x_{k}-\ell \vert \geq \varepsilon \} )$$ has natural density zero for $$\varepsilon >0$$. is necessarily unique, which is statistical limit of $$( x_{k} )$$, and written as $${S\mbox{-}\lim x_{k}=\ell }$$. The space of all statistically convergent sequences is denoted by S (see ).

The order of statistical convergence of a sequence of positive linear operators was given by Gadjiev and Orhan , and after that Çolak  introduced statistical convergence of order α and strong p-Cesàro summability of order α.

Statistical convergence was introduced for double sequences by Mursaleen and Edely . Besides this topic was studied by many authors (such as [15, 24, 25]). For some further works in this direction, we refer to .

The concepts of convergence and statistical convergence for double sequence can be expressed as follows.

Let $$s^{2}$$ denote the space of all double sequences, and let $$\ell_{\infty }^{2}$$, $$c^{2}$$ and $$c_{0}^{2}$$ be the linear spaces of bounded, convergent and null sequences $$x= ( x_{jk} )$$ with complex terms, respectively, normed by $$\Vert x\Vert _{ ( \infty,2 ) }=\sup_{j,k}\vert x_{jk}\vert$$, where j, $$k\in \mathbb{N}= \{ 1,2,\, \ldots \}$$.

A double sequence $$x= ( x_{j,k} ) _{j,k=0}^{\infty }$$ has Pringsheim limit provided that for every $$\varepsilon >0$$ there exists $$N\in \mathbb{N}$$ such that $$\vert x_{j,k}-\ell \vert < \varepsilon$$ whenever $$j,k>N$$. In this case, we write $$P\mbox{-}\lim x=\ell$$ .

$$x= ( x_{j,k} ) _{j,k=0}^{\infty }$$ is bounded if there exists a positive number M such that $$\vert x_{j,k}\vert < M$$ for all j and k, that is, $$\Vert x\Vert =\sup_{j,k\geq 0}\vert x_{j,k}\vert <\infty$$.

Let $$K\subseteq \mathbb{N} \times \mathbb{N}$$ and $$K ( m,n ) = \{ ( j,k ) :j\leq m,k\leq n \}$$. The double natural density of K is defined by

$$\delta_{2} ( K ) =P\mbox{-}\lim_{m,n}\frac{1}{mn}\bigl\vert K ( m,n ) \bigr\vert \quad \mbox{if the limit exists.}$$

A double sequence $$x= ( x_{jk} ) _{j,k\in \mathbb{N}}$$ is said to be statistically convergent to if for every $$\varepsilon >0$$ the set $$\{ ( j,k ) :j\leq m,k\leq n:\vert x_{jk}- \ell \vert \geq \varepsilon \}$$ has double natural density zero . In this case, one can write $$st_{2}\mbox{-}\lim x=\ell$$, and we denote the collection of all statistically convergent double sequences by $$st_{2}$$. Recently, Çolak and Altin  introduced double statistically convergent of order α, and they examined some inclusion relations.

The idea of a modulus function was introduced in 1953 by Nakano . Later, Ruckle  and Maddox  used this concept to construct some sequence spaces. Let us remind modulus function.

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

1. 1.

$$f ( x ) =0$$ if and only if $$x=0$$,

2. 2.

$$f ( x+y ) \leq f ( x)+f(y )$$ for every $$x,y\in \mathbb{R}^{+}$$,

3. 3.

f is increasing,

4. 4.

f is continuous from the right at 0.

Hence, f must be continuous everywhere on $$[ 0,\infty )$$. A modulus function may be bounded or unbounded. For example, $$f ( x ) =\frac{x}{1+x}$$ is bounded, but $$f ( x ) =x ^{p}$$, $$0< p\leq 1$$ is unbounded.

Aizpuru et al.  introduced and discussed the concepts of f-statistical convergence and f-statistically Cauchy sequences, a single sequence of numbers, where f is an unbounded modulus function. Bhardwaj and Dhawan  continued this work and defined f-statistical convergence of order α. This new idea was introduced by Borgohain and Savaş  under the name of ’$$f_{\lambda }$$-statistical convergence’. Aizpuru et al. also studied these concepts for double sequences . Mursaleen  introduced λ-statistical convergence as an extension of $$( V, \lambda )$$-summability of Leindler  with the help of a non-decreasing sequence, $$\lambda = ( \lambda_{n} )$$ being a non-decreasing sequence of positive numbers tending to ∞ with $$\lambda_{n+1}\leq \lambda_{n}+1$$, $$\lambda _{1}=1$$. The generalized de la Vallee-Poussin mean is defined by

$$t_{n} ( x ) =\frac{1}{\lambda_{n}}{\sum_{k\in I_{n}}} x_{k},$$

where $$I_{n}= [ n-\lambda_{n}+1,n ]$$.

λ-statistical convergence of double sequences has been expressed by Mursaleen et al. .

## $$f_{\lambda,\mu }$$-double statistical convergence of order α̃

In this section, we introduce $$f_{\lambda,\mu }$$-double statistical convergence of order α̃ for double sequences.

Throughout this paper, we take $$s,t,u,v\in ( 0,1 ]$$ as otherwise indicated. We will write α̃ instead of $$( s,t )$$ and β̃ instead of $$( u,v )$$. Also, we define the following:

\begin{aligned}& \widetilde{\alpha } \preceq \widetilde{\beta }\quad \Longleftrightarrow \quad s \leq u\quad \mbox{and}\quad t\leq v, \\& \widetilde{\alpha } \prec \widetilde{\beta }\quad \Longleftrightarrow \quad s< u\quad \mbox{and}\quad t< v, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\& \widetilde{\alpha } \cong \widetilde{\beta }\quad \Longleftrightarrow \quad s=u\quad \mbox{and}\quad t=v, \\& \widetilde{\alpha } \in ( 0,1 ] \quad \Longleftrightarrow \quad s, t \in ( 0,1 ], \\& \widetilde{\beta } \in ( 0,1 ] \quad \Longleftrightarrow \quad u, v \in ( 0,1 ], \\& \widetilde{\alpha } \cong 1\quad \mbox{in case }s=t=1, \\& \widetilde{\beta } \cong 1\quad \mbox{in case }u=v=1, \\& \widetilde{\alpha } \succ 1\quad \mbox{in case }s>1\quad \mbox{and}\quad t>1. \end{aligned}

Furthermore, we write $$S_{\tilde{\alpha }}^{2} ( f,\lambda, \mu )$$ to denote $$S_{ ( s,t ) }^{2} ( f,\lambda,\mu )$$ and $$S_{\widetilde{\beta }}^{2}( f,\lambda,\mu )$$ to denote $$S_{ ( u,v ) }^{2} ( f,\lambda, \mu )$$ in the section below.

We begin with the following definitions.

Let $$\lambda = ( \lambda_{n} )$$ and $$\mu = ( \mu_{m} )$$ be two non-decreasing sequences of positive real numbers tending to ∞ with $$\lambda_{n+1}\leq \lambda_{n}+1$$, $$\lambda_{1}=0; \mu_{n+1}\leq \mu_{n}+1$$, $$\mu_{1}=0$$ and $$\widetilde{\alpha }\in ( 0,1 ]$$ be given.

Let $$K\subseteq \mathbb{N}\times \mathbb{N}$$ be a two-dimensional set of positive integers and f be an unbounded modulus function. Then $$\delta_{\tilde{\alpha }}^{{f}2} ( {\lambda,\mu } )$$-double $$density$$ of K is defined as

$$\delta_{\tilde{\alpha }}^{{f}2}(K)=\lim_{n,m\rightarrow \infty } \frac{1}{f ( \lambda_{n}^{s}\mu_{m}^{t} ) }f \bigl( \bigl\vert \bigl\{ (j,k) \in I_{n}\times I_{m}: ( i,j ) \in K\bigr\} \bigr\vert \bigr) \quad \mbox{if the limit exists.}$$

### Definition 2.1

Let $$\lambda = ( \lambda_{n} )$$ and $$\mu = ( \mu_{m} )$$ be two non-decreasing sequences of positive real numbers as above and $$\widetilde{\alpha }\in ( 0,1 ]$$ be given.

$$( x_{jk} )$$ is said to be $$f_{\lambda,\mu }$$-statistically convergent of order α̃ if there is a complex number such that, for every $$\varepsilon >0$$,

$$\lim_{n,m\rightarrow \infty }\frac{1}{f ( \lambda_{n}^{s}\mu_{m} ^{t} ) }f \bigl( \bigl\vert \bigl\{ (j,k) \in I_{n}\times I_{m}: \vert x_{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert \bigr) =0.$$

In this case we write $$S_{\tilde{\alpha }}^{2} ( f,\lambda, \mu ) \mbox{-}\lim_{j,k}x_{jk}=\ell$$, and we denote the set of all $$f_{\lambda,\mu }$$-statistically convergent double sequences of order α̃ by $$S_{\tilde{\alpha }}^{2} ( f, \lambda,\mu )$$, where f is an unbounded modulus function.

In the case of $$f(x)=x$$, $$\widetilde{\alpha }\cong 1$$ and $$\lambda _{n}=n$$, $$\mu_{m}=m$$, $$f_{\lambda,\mu }$$-statistical convergence of order α̃ reduces to the statistical convergence of double sequences . If $$x= ( x_{jk} )$$ is $$f_{\lambda,\mu }$$-statistically convergent of order α̃ to the number , then is determined uniquely. $$f_{\lambda,\mu }$$-double statistical convergence of order α̃ is well defined for $$\widetilde{\alpha }\in ( 0,1 ]$$ but it is not well defined for $$\widetilde{\alpha }\succ 1$$. For this, let us define $$x= ( x_{jk} )$$ as follows:

$$x_{jk}= \textstyle\begin{cases} 1,&\mbox{if }j+k \mbox{ even}, \\ 0,&\mbox{if }j+k\mbox{ odd}. \end{cases}$$

Since $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$, we have

\begin{aligned} & \lim_{n,m\rightarrow \infty }\frac{1}{f ( \lambda_{n}^{s}\mu _{m}^{t} ) }f \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n}\times I_{m}:\vert x_{jk}-1\vert \geq \varepsilon \bigr\} \bigr\vert \bigr) \leq \lim_{n,m\rightarrow \infty }\frac{f ( [ \vert \lambda _{n}^{s}\mu_{m}^{t}\vert ] ) +1}{f ( 2\lambda _{n}^{s}\mu_{m}^{t} ) }=0 \end{aligned}

and

\begin{aligned} & \lim_{n,m\rightarrow \infty }\frac{1}{f ( \lambda_{n}^{s}\mu _{m}^{t} ) }\bigl\vert \bigl\{ (j,k)\in I_{n}\times I_{m} : \vert x_{jk}-0 \vert \geq \varepsilon \bigr\} \bigr\vert \leq \lim_{n,m\rightarrow \infty }\frac{f ( [ \vert \lambda _{n}^{s}\mu_{m}^{t}\vert ] ) +1}{f ( 2\lambda _{n}^{s}\mu_{m}^{t} ) }=0 \end{aligned}

for $$\widetilde{\alpha }\succ 1$$, that is, $$s>1$$ and $$t>1$$, so that $$x= ( x_{jk} )$$ is $$f_{\lambda,\mu }$$-statistically convergent of order α̃ both to 1 and 0, i.e., $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-}\lim x_{jk}=1$$ and $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-}\lim x_{jk}=0$$. But this is impossible.

### Theorem 2.2

Let f be an unbounded modulus function and $$\widetilde{\alpha }\in ( 0,1 ]$$. Let $$x= ( x_{jk} )$$, $$y= ( y_{jk} )$$ be any two sequences of complex numbers. Then

1. (i)

If $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-}\lim x _{jk}=\ell_{0}$$ and $$c\in \mathbb{C}$$, then $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-}\lim cx_{jk}=c\ell_{0}$$;

2. (ii)

If $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-} \lim x_{jk}=\ell_{o}$$ and $$S_{\tilde{\alpha }}^{2} ( f,\lambda, \mu ) \mbox{-}\lim y_{jk}=\ell_{1}$$, then $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \mbox{-}\lim ( x_{jk}+y_{jk} ) =\ell _{0}+\ell_{1}$$.

### Theorem 2.3

Let f be an unbounded modulus function and $$\tilde{\alpha },\tilde{\beta }$$ be two real numbers such that 0 $$\widetilde{\alpha }\preceq \widetilde{\beta }\preceq 1$$. Then $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \subseteq S_{\tilde{\beta }}^{2} ( f,\lambda,\mu )$$ and strict inclusion may occur.

### Proof

Let $$\tilde{\alpha },\tilde{\beta }\in (0,1]$$ be given such that $$\tilde{\alpha }\leq \tilde{\beta }$$. Since f is increasing, we have

\begin{aligned} &\frac{1}{f ( \lambda_{n}^{u}\mu_{m}^{v} ) } f \bigl( \bigl\vert \bigl\{ (j,k) \in I_{n} \times I_{m}:\vert x_{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert \bigr) \\ & \quad \leq \frac{1}{f ( \lambda_{n}^{s}\mu_{m}^{t} ) }f \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n} \times I_{m}:\vert x_{jk}- \ell \vert \geq \varepsilon \bigr\} \bigr\vert \bigr) \end{aligned}

for every $$\varepsilon >0$$, and this gives $$S_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \subseteq S_{\tilde{\beta }}^{2} ( f, \lambda,\mu )$$. To show that the strict inclusion may occur, consider a sequence $$x= ( x_{jk} )$$ defined by

$$x_{jk=} \textstyle\begin{cases} jk, & \mbox{if }n- [ \vert \lambda_{n}\vert ] +1 \leq j\leq n\mbox{ and }m- [ \vert \mu_{m}\vert ] +1\leq k\leq m, \\ 0, & \mbox{otherwise} \end{cases}$$

and we take $$f ( x ) =x^{p}$$, $$( 0< p\leq 1 )$$ and hence $$x\in S_{\tilde{\beta }}^{2} ( f,\lambda,\mu )$$ for $$\tilde{\beta }\in (\frac{1}{2},1]$$, (i.e., $$\frac{1}{2}< u\leq 1$$ and $$\frac{1}{2}< v\leq 1$$ ), but $$x\notin S_{\tilde{\alpha }}^{2} ( f, \lambda,\mu )$$ for α̃ $$\in (0,\frac{1}{2}]$$ (i.e., $$0< s\leq \frac{1}{2}$$ and $$0< t\leq \frac{1}{2}$$ ). □

The following results can be easily derived from Theorem 2.3.

### Corollary 2.4

If $$x= ( x_{jk} )$$ is $$f_{\lambda, \mu }$$-statistically convergent of order α̃ to , for some α̃ such that $$\widetilde{\alpha }\in ( 0,1 ]$$, then it is $$f_{\lambda, \mu }$$-statistically convergent to , and the inclusion is strict.

### Corollary 2.5

Let $$\widetilde{\alpha },\widetilde{\beta } \in ( 0,1 ]$$ be given. Then

1. (i)

$$S_{\widetilde{\alpha }}^{2} ( f,\lambda,\mu ) =S_{ \widetilde{\beta }}^{2} ( f,\lambda,\mu )$$ if $$\widetilde{\alpha }\cong \widetilde{\beta }$$.

2. (ii)

$$S_{\widetilde{\alpha }}^{2} ( f,\lambda,\mu ) =S ^{2} ( f,\lambda,\mu )$$ if $$\widetilde{\alpha }\cong 1$$.

## Strongly double Cesàro summability of order α̃ defined by a modulus function

In this section, we give the relationships between the spaces $$w_{\tilde{\alpha },0}^{2} ( f,\lambda,\mu ) ,w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu )$$ and $$w_{ \tilde{\alpha },\infty }^{2} ( f,\lambda,\mu )$$.

### Definition 3.1

Let f be a modulus function and α̃ be a positive real number. We have

\begin{aligned}& w_{\tilde{\alpha },0}^{2} ( f,\lambda,\mu ) = \biggl\{ x=(x _{jk})\in s^{2}:\lim_{n,m\rightarrow \infty } \frac{1}{ ( \lambda _{n}\mu_{m} ) ^{\tilde{\alpha }}}{\sum_{j\in J_{n}}} {\sum _{k\in I_{n}}} f \bigl( \vert x_{jk}\vert \bigr) =0 \biggr\} , \\& w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) = \biggl\{ x=(x _{jk})\in s^{2}:\lim_{n,m\rightarrow \infty } \frac{1}{ ( \lambda _{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} { \sum _{k\in I_{n}}} f \bigl( \vert x_{jk}-\ell \vert \bigr) =0 \biggr\} , \\& w_{\tilde{\alpha },\infty }^{2} ( f,\lambda,\mu ) = \biggl\{ x=(x_{jk})\in s^{2}:\sup_{n,m} \frac{1}{ ( \lambda_{n}\mu _{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} {\sum _{k\in I_{n}}} f \bigl( \vert x_{jk}\vert \bigr) < \infty \biggr\} . \end{aligned}

### Theorem 3.2

1. (i)

Let f be a modulus function. For $$\tilde{\alpha }\succ 0$$, we have $$w_{\tilde{\alpha },0}^{2} ( f, \lambda,\mu ) \subset$$ $$w_{\tilde{\alpha },\infty }^{2} ( f,\lambda,\mu )$$.

2. (ii)

Let f be a modulus function. For α̃ 1, we have $$w_{\tilde{\alpha }}^{2} ( f, \lambda,\mu ) \subset$$ $$w_{\tilde{\alpha },\infty }^{2} ( f,\lambda,\mu )$$.

### Proof

(i) The proof of (i) is trivial.

(ii) Let $$x\in w_{\tilde{\alpha }}^{2} ( f,\lambda, \mu )$$. By the definition of modulus function (ii) and (iii), we have

\begin{aligned}& \frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} {\sum _{k\in I_{n}}} f \bigl( \vert x_{jk}\vert \bigr) \leq \frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}} {\sum_{j\in J_{n}}} {\sum _{k\in I_{n}}} f \bigl( \vert x_{jk}-\ell \vert \bigr) +f \bigl( \vert \ell \vert \bigr) \frac{1}{ ( \lambda_{n}\mu _{m} ) ^{\tilde{\alpha }}}{ \sum _{j\in J_{n}}} {\sum_{k\in I_{n}}} 1, \end{aligned}

and since α̃ 1 and $$x\in w_{\tilde{\alpha }} ^{2} ( f,\lambda,\mu )$$, we have $$x\in w_{ \tilde{\alpha },\infty }^{2} ( f,\lambda,\mu )$$, which completes the proof. □

### Theorem 3.3

For any modulus function f and α̃ 1, we have $$w_{\tilde{\alpha }}^{2} ( \lambda,\mu ) \subset$$ $$w_{\tilde{\alpha }}^{2} ( f, \lambda,\mu )$$, $$w_{\tilde{\alpha },0}^{2} ( \lambda, \mu ) \subset$$ $$w_{\tilde{\alpha },0}^{2} ( f,\lambda, \mu )$$ and $$w_{\tilde{\alpha },\infty }^{2} ( \lambda, \mu ) \subset$$ $$w_{\tilde{\alpha },\infty }^{2} ( f,\lambda,\mu )$$.

### Proof

We give the proof only when $$w_{\tilde{\alpha }, \infty }^{2} ( \lambda,\mu ) \subset$$ $$w_{\tilde{\alpha }, \infty }^{2} ( f,\lambda,\mu )$$ and the rest of cases will follow similarly. Let $$x\in w_{\tilde{\alpha },\infty }^{2} ( \lambda,\mu )$$, so that

$$\sup_{n,m} \frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum _{j\in J_{n}}} {\sum_{k\in I_{n}}} \vert x_{jk}\vert < \infty.$$

Let $$\varepsilon >0$$ and choose δ with $$0<\delta <1$$ such that $$f ( t ) <\varepsilon$$ for $$0\leq t<\delta$$. Now we write

$$\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} { \sum _{k\in I_{n}}} f \bigl( \vert x_{jk}\vert \bigr) =\sum _{1}+\sum_{2},$$

where the first summation is over $$\vert x_{jk}\vert \leq \delta$$ and the second is over $$\vert x_{jk}\vert > \delta$$. Then $$\sum_{1}\leq \varepsilon\cdot\frac{1}{ ( \lambda_{n} \mu_{m} ) ^{\tilde{\alpha }-1}}$$ and, for $$\vert x_{jk}\vert > \delta$$, we use the fact that

$$\vert x_{jk}\vert < \frac{\vert x_{jk}\vert }{ \delta }< 1+ \biggl[ \biggl\vert \frac{\vert x_{jk}\vert }{ \delta }\biggr\vert \biggr],$$

where $$[ \vert t\vert ]$$ denotes the integer part of t. Given $$\varepsilon >0$$, by the definition of f, we have

$$f \bigl( \vert x_{jk}\vert \bigr) \leq \biggl( 1+ \biggl[ \biggl\vert \frac{\vert x_{jk}\vert }{\delta }\biggr\vert \biggr] \biggr) f ( 1 ) \leq 2f ( 1 ) \frac{\vert x _{jk}\vert }{\delta }$$

for $$\vert x_{jk}\vert >\delta$$ and hence $$\sum_{2} \leq 2f ( 1 ) \delta^{-1}{ \sum_{j\in J_{n}}} { \sum_{k\in I_{n}}} \vert x_{jk}\vert$$, which together with $$\sum_{1}\leq \varepsilon \frac{1}{ ( \lambda_{n}\mu _{m} ) ^{\tilde{\alpha }-1}}$$ yields

$$\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} { \sum _{k\in I_{n}}} f \bigl( \vert x_{jk}\vert \bigr) \leq \varepsilon \cdot\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{ \tilde{\alpha }-1}}+2f ( 1 ) \delta^{-1}\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} { \sum_{k\in I_{n}}} \vert x_{jk}\vert .$$

Since $$\tilde{\alpha }\geq 1$$ and $$x\in w_{\tilde{\alpha },\infty } ^{2} ( \lambda,\mu )$$, we have $$x\in w_{\tilde{\alpha }, \infty }^{2} ( f,\lambda,\mu )$$ and the proof is complete. □

### Theorem 3.4

Let f be a modulus function f and $$\tilde{\alpha }\succ 0$$. If $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$, then $$w_{\tilde{\alpha }}^{2} ( f, \lambda,\mu ) \subset w_{\tilde{\alpha }}^{2}( \lambda,\mu )$$.

### Proof

Following the proof of Proposition 1 of Maddox , we have $$l=\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}= \inf \{ \frac{f ( t ) }{t}:t>0 \}$$. By the definition of l, we have $$f ( t ) \geq lt$$ for all $$t\geq 0$$. Since $$l>0$$, we get $$t\leq l^{-1}f ( t )$$ for all $$t\geq 0$$, and so

$$\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}}{ \sum_{j\in J_{n}}} { \sum _{k\in I_{n}}} \vert x_{jk}-\ell \vert \leq l ^{-1}\frac{1}{ ( \lambda_{n}\mu_{m} ) ^{\tilde{\alpha }}} { \sum_{j\in J_{n}}} { \sum_{k\in I_{n}}} f \bigl( \vert x_{jk}-\ell \vert \bigr),$$

from where it follows that $$x\in w_{\tilde{\alpha }}^{2} ( f, \lambda,\mu )$$ whenever $$x\in w_{\tilde{\alpha }}^{2} ( \lambda,\mu )$$. □

### Theorem 3.5

For any modulus f such that $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$ and α̃ 1. Then $$w_{\tilde{\alpha }}^{2} ( \lambda,\mu ) =w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu )$$.

## Relation between $$f_{\lambda,\mu }$$-statistical convergence of order α̃ and strongly double Cesàro summability of order α̃ defined by a modulus function

In this section, we give the relationship between the strong $$f_{\lambda,\mu }$$-Cesàro summability of order α̃ and $$f_{\lambda,\mu }$$-statistical convergence of order β̃.

### Lemma 4.1

Let f be an unbounded function such that there is a positive constant c such that $$f ( xy ) \geq cf ( x ) f ( y )$$ for all $$x\geq 0$$, $$y\geq 0$$ .

### Theorem 4.2

Let $$0\prec \tilde{\alpha }\preceq \tilde{\beta }\preceq 1$$ and f be an unbounded modulus function such that there is a positive constant c such that $$f ( xy ) \geq cf ( x ) f ( y )$$ for all $$x\geq 0$$, $$y\geq 0$$ and $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$. If a sequence $$x=(x_{jk})$$ is strongly $$f_{\lambda,\mu }$$-Cesàro summable of order α̃ with respect to f to , then it is $$f_{\lambda,\mu }$$-statistically convergent of order β̃ to .

### Proof

For any sequence $$x=(x_{jk})$$ and $$\varepsilon >0$$, using the definition of modulus function (ii) and (iii), we have

\begin{aligned} { \sum_{j\in J_{n}}} { \sum_{k\in I_{n}}} f \bigl( \vert x_{jk}-L\vert \bigr) & \geq f \biggl( { \sum _{j\in J_{n}}} { \sum_{k\in I_{n}}} \vert x_{jk}-\ell \vert \biggr)\geq f \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n}\times I_{m} : \vert x_{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert \varepsilon \bigr) \\ & \geq cf \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n}\times I_{m}:\vert x _{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert f( \varepsilon ) \bigr) \end{aligned}

and since α̃ β̃

\begin{aligned} & \frac{1}{n^{s}m^{t}}{ \sum_{j=1}^{n}} { \sum_{k=1}^{m}} f \bigl( \vert x_{jk}-\ell \vert \bigr) \\ & \quad \geq \frac{1}{n^{s}m^{t}}cf \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n} \times I_{m} : \vert x_{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert f ( \varepsilon ) \bigr) \\ &\quad \geq \frac{1}{n^{u}m^{v}}cf \bigl( \bigl\vert \bigl\{ (j,k)\in I_{n} \times I_{m} :\vert x_{jk}-\ell \vert \geq \varepsilon \bigr\} \bigr\vert f ( \varepsilon ) \bigr) \\ & \quad =\frac{1}{n^{u}m^{v}f ( n^{u}m^{v} ) }cf \bigl( \bigl\vert \bigl\{ (j,k) \in I_{n}\times I_{m} :\vert x_{jk}-\ell \vert \geq \varepsilon\bigr\} \bigr\vert f ( \varepsilon ) \bigr) f \bigl( n^{u}m^{v} \bigr), \end{aligned}

where, using the fact that $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$ and $$x\in w_{\tilde{\alpha }}^{2} ( f, \lambda,\mu )$$, it follows that $$x\in S_{\tilde{\beta }}^{2} ( \lambda,\mu )$$ and the proof is complete. □

If we take β̃ α̃ in Theorem 4.2, we have the following.

### Corollary 4.3

Let f be an unbounded modulus function $$f ( xy ) \geq cf ( x ) f ( y )$$, where c is a positive constant for all $$x\geq 0$$, $$y\geq 0$$ and $$\lim_{t\rightarrow \infty }\frac{f ( t ) }{t}>0$$ and $$\tilde{\alpha }\in (0,1]$$. If a sequence is strongly $$f_{\lambda, \mu }$$-Cesàro summable of order α̃ with respect to f to , then it is $$f_{\lambda,\mu }$$-statistically convergent of order α̃ to .

## Conclusions

In this study, we define $$f_{\lambda,\mu }$$-statistical convergence for double sequences of order α̃, where f is an unbounded modulus function. Besides this we also study strong $$f_{\lambda,\mu }$$-Cesàro summability for double sequences of order α̃ and give inclusion relations. These results are the generalizations of the studies by Meenakshi et al. .

## References

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