Open Access

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

Journal of Inequalities and Applications20172017:246

https://doi.org/10.1186/s13660-017-1512-y

Received: 21 June 2017

Accepted: 12 September 2017

Published: 4 October 2017

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.

Keywords

double sequencesstatistical convergenceCesàro summability

MSC

40A0540C0546A45

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 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 [520]).

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 [2630].

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 \(P\mbox{-}\lim x=\ell \) [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 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 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 , 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 \(\widetilde{\alpha }\cong 1\).

     

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 , 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 .

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].

Declarations

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Education, Harran University
(2)
Department of Mathematics, Fırat University

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