1 Introduction

The first idea of statistical convergence goes back to the first edition of the famous Zygmund’s monograph [1]. The statistical convergence was introduced for real and complex sequences by Steinhaus [2]. Fast [3] extended the usual concept of sequential limit and called it statistical convergence. Schoenberg [4] 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 [5]). A sequence x= \(( x_{k} ) \) of complex numbers is said to be statistically convergent to some number ℓ if \(\delta ( \{ k\in \mathbb{N}:\vert x_{k}-\ell \vert \geq \varepsilon \} ) \) has natural density zero for \(\varepsilon >0\). ℓ 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 [5–20]).

The order of statistical convergence of a sequence of positive linear operators was given by Gadjiev and Orhan [21], and after that Çolak [22] 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 [23]. Besides this topic was studied by many authors (such as [15, 24, 25]). For some further works in this direction, we refer to [26–30].

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 ℓ 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 \(P\mbox{-}\lim x=\ell \) [31].

\(x= ( x_{j,k} ) _{j,k=0}^{\infty }\) 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 ℓ 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 [23]. 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 [27] 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 [32]. Later, Ruckle [33] and Maddox [34] 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. [35] 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 [36] continued this work and defined f-statistical convergence of order α. This new idea was introduced by Borgohain and Savaş [37] under the name of ’\(f_{\lambda }\)-statistical convergence’. Aizpuru et al. also studied these concepts for double sequences [38]. Mursaleen [39] introduced λ-statistical convergence as an extension of \(( V, \lambda ) \)-summability of Leindler [40] 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. [41].

2 \(f_{\lambda,\mu }\)-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 ℓ 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 [23]. If \(x= ( x_{jk} ) \) is \(f_{\lambda,\mu }\)-statistically convergent of order α̃ to the number ℓ, then ℓ 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 ℓ, for some α̃ such that \(\widetilde{\alpha }\in ( 0,1 ] \), then it is \(f_{\lambda, \mu }\)-statistically convergent to ℓ, 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 \(\widetilde{\alpha }\cong 1\).

3 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 [42], 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 \(w_{\tilde{\alpha }}^{2} ( \lambda,\mu ) =w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) \).

4 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 \(x\geq 0\), \(y\geq 0\) [42].

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 ℓ, then it is \(f_{\lambda,\mu }\)-statistically convergent of order β̃ to ℓ.

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 ℓ, then it is \(f_{\lambda,\mu }\)-statistically convergent of order α̃ to ℓ.

5 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. [43].