# On generalized difference lacunary statistical convergence in a paranormed space

## Abstract

In this article, we introduce the concept of ${\mathrm{\Delta }}^{m}$-lacunary statistical convergence and ${\mathrm{\Delta }}^{m}$-lacunary strong convergence in a paranormed space. Also, we establish some connections between these concepts.

## Dedication

Dedicated to Professor Hari M Srivastava.

## 1 Introduction

In order to extend convergence of sequences, the notion of statistical convergence was introduced by Fast [1] and Steinhaus [2] and several generalizations and applications of this concept have been investigated by various authors [3, 4]. This notion was studied in normed spaces by Kolk [5], in locally convex Hausdorff topological spaces by Maddox [6], in topological Hausdorff groups by Çakallı [7] and in probabilistic normed space by Karakuş [8]. Recently, Alotaibi and Alroqi [9] extended this notion in paranormed spaces.

In this article, we study the concept of statistical convergence from difference sequence spaces which are defined over paranormed space.

## 2 Preliminaries and definitions

Let K be a subset of the set of natural numbers . Then the asymptotic density of K denoted by $\delta \left(K\right)={lim}_{n}\frac{1}{n}|\left\{k\le n:k\in K\right\}|$, where the vertical bars denote the cardinality of the enclosed set in [10].

A number sequence $x=\left({x}_{k}\right)$ is said to be statistically convergent to the number L if for each $\epsilon >0$, the set $K\left(\epsilon \right)=\left\{k\le n:|{x}_{k}-L|\ge \epsilon \right\}$ has asymptotic density zero, i.e.,

$\underset{n}{lim}\frac{1}{n}|\left\{k\le n:|{x}_{k}-L|\ge \epsilon \right\}|=0.$

In this case we write $st\text{-}limx=L$. This concept was studied by [11, 12].

By a lacunary $\theta =\left({k}_{r}\right)$; $r=0,1,2,\dots$ , where ${k}_{0}=0$, we shall mean an increasing sequence of non-negative integers with ${k}_{r}-{k}_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by ${I}_{r}=\left({k}_{r-1},{k}_{r}\right]$ and ${h}_{r}={k}_{r}-{k}_{r-1}$. The ratio $\frac{{k}_{r}}{{k}_{r-1}}$ will be denoted by ${q}_{r}$.

The notion of difference sequence space $X\left(\mathrm{\Delta }\right)$ was introduced by Kızmaz [13] as follows:

$X\left(\mathrm{\Delta }\right)=\left\{x=\left({x}_{k}\right):\left(\mathrm{\Delta }{x}_{k}\right)\in X\right\}$

for $X={l}_{\mathrm{\infty }}$, c, ${c}_{0}$, where $\mathrm{\Delta }{x}_{k}={x}_{k}-{x}_{k+1}$ for all $k\in \mathbb{N}$.

The notion of difference sequence spaces was further generalized by Et and Çolak [14] as follows:

$X\left({\mathrm{\Delta }}^{m}\right)=\left\{x=\left({x}_{k}\right)\in w:\left({\mathrm{\Delta }}^{m}{x}_{k}\right)\in X\right\}$

for $X={l}_{\mathrm{\infty }}$, c and ${c}_{0}$, where $m\in \mathbb{N}$, ${\mathrm{\Delta }}^{m}{x}_{k}={\mathrm{\Delta }}^{m-1}{x}_{k}-{\mathrm{\Delta }}^{m-1}{x}_{k+1}$, ${\mathrm{\Delta }}^{0}{x}_{k}={x}_{k}$.

The sequence x is said to be ${\mathrm{\Delta }}^{m}$-statistically convergent to the number L provided that for each $\epsilon >0$,

$\underset{n}{lim}\frac{1}{n}|\left\{k\le n:|{\mathrm{\Delta }}^{m}{x}_{k}-L|\ge \epsilon \right\}|=0.$

The set of all ${\mathrm{\Delta }}^{m}$-statistically convergent sequences was denoted by $S\left({\mathrm{\Delta }}^{m}\right)$ in [15].

Furthermore, this notion was studied in [16, 17].

A paranorm is a function $g:X⟶\mathbb{R}$ defined on a linear space X such that for all $x,y,z\in X$,

(i) $g\left(x\right)=0$ if $x=\theta$;

(ii) $g\left(-x\right)=g\left(x\right)$;

(iii) $g\left(x+y\right)\le g\left(x\right)+g\left(y\right)$;

(iv) If $\left({\alpha }_{n}\right)$ is a sequence of scalars with ${\alpha }_{n}⟶{\alpha }_{0}$ ($n⟶\mathrm{\infty }$) and ${x}_{n},a\in X$ with ${x}_{n}⟶a$ ($n⟶\mathrm{\infty }$) in the sense that $g\left({x}_{n}-a\right)⟶0$ ($n⟶\mathrm{\infty }$), then ${\alpha }_{n}{x}_{n}⟶{\alpha }_{0}a$ ($n⟶\mathrm{\infty }$), in the sense that $g\left({\alpha }_{n}{x}_{n}-{\alpha }_{0}a\right)⟶0$ ($n⟶\mathrm{\infty }$).

A paranorm g for which $g\left(x\right)=0$ implies $x=\theta$ is called a total paranorm on X and the pair $\left(X,g\right)$ is called a total paranormed space.

Note that each seminorm (norm) p on X is a paranorm (total) but converse need not be true.

The concept of paranorm is a generalization of absolute value [18].

A modulus function f is a function from $\left[0,\mathrm{\infty }\right)$ to $\left[0,\mathrm{\infty }\right)$ such that

(i) $f\left(x\right)=0$ if and only if $x=0$;

(ii) $f\left(x+y\right)\le f\left(x\right)+f\left(y\right)$ for all $x,y\ge 0$;

(iii) f increasing;

(iv) f is continuous from at the right zero.

Since $|f\left(x\right)-f\left(y\right)|\le f\left(|x-y|\right)$, it follows from condition (iv) that f is continuous on $\left[0,\mathrm{\infty }\right)$. Furthermore, we have $f\left(nx\right)\le nf\left(x\right)$ for all $n\in \mathbb{N}$ from condition (ii) and so

$f\left(x\right)=f\left(nx\frac{1}{n}\right)\le nf\left(\frac{x}{n}\right).$

Hence, for all $n\in \mathbb{N}$,

$\frac{1}{n}f\left(x\right)\le f\left(\frac{x}{n}\right).$

A modulus may be bounded or unbounded. For example, $f\left(x\right)={x}^{p}$ for $0 is unbounded, but $f\left(x\right)=\frac{x}{1+x}$ is bounded. Ruckle [19] used the idea of a modulus function f to construct a class of FK spaces

$L\left(f\right)=\left\{x=\left({x}_{k}\right):\sum f\left(|{x}_{k}|\right)<\mathrm{\infty }\right\}.$

In [9], the notion of statistical convergence was defined in a paranormed space.

Definition 2.1 A sequence $x=\left({x}_{k}\right)$ is said to be statistically convergent to the number L in $\left(X,g\right)$ if for each $\epsilon >0$,

$\underset{n}{lim}\frac{1}{n}|\left\{k\le n:g\left({x}_{k}-L\right)\ge \epsilon \right\}|=0.$

In this case, we write $g\left(st\right)\text{-}limx=L$. We denote the set of all $g\left(st\right)$-convergent sequences by ${S}_{g}$ [9].

Definition 2.2 A sequence $x=\left({x}_{k}\right)$ is said to be strongly p-Cesaro summable ($0) to the limit L in $\left(X,g\right)$ if

$\underset{n}{lim}\frac{1}{n}\sum _{j=1}^{n}{\left(g\left({x}_{j}-L\right)\right)}^{p}=0,$

and we write it as ${x}_{k}⟶L\left({\left[{C}_{1},g\right]}_{p}\right)$. In this case, L is called the ${\left[{C}_{1},g\right]}_{p}\text{-}lim$ it of x [9].

In this article, we shall study the concept of ${\mathrm{\Delta }}^{m}$-lacunary statistical convergence, ${\mathrm{\Delta }}^{m}$-lacunary strong convergence and ${\mathrm{\Delta }}^{m}$-lacunary strong convergence with respect to a modulus function in a paranormed space.

## 3 Generalized difference statistical convergence in a paranormed space

Definition 3.1 A sequence $x=\left({x}_{k}\right)$ is said to be ${\mathrm{\Delta }}^{m}$-statistically convergent to the number L in $\left(X,g\right)$ if for each $\epsilon >0$,

$\underset{n}{lim}\frac{1}{n}|\left\{k\le n:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon \right\}|=0.$

In this case, we write ${S}_{g}\left({\mathrm{\Delta }}^{m}\right)\text{-}limx=L$. We denote the set of all ${\mathrm{\Delta }}^{m}$-statistically convergent sequences in $\left(X,g\right)$ by ${S}_{g}\left({\mathrm{\Delta }}^{m}\right)$.

Definition 3.2 Let θ be a lacunary sequence. A sequence $x=\left({x}_{k}\right)$ is said to be ${\mathrm{\Delta }}^{m}$-lacunary statistically convergent to the number L in $\left(X,g\right)$ if for each $\epsilon >0$,

$\underset{r}{lim}\frac{1}{{h}_{r}}|\left\{k\in {I}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon \right\}|=0.$

In this case, we write ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\text{-}limx=L$. We denote the set of all ${\mathrm{\Delta }}^{m}$-lacunary statistically convergent sequences in $\left(X,g\right)$ by ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$.

Definition 3.3 A sequence $x=\left({x}_{k}\right)$ is said to be strongly ${\mathrm{\Delta }}^{m}$-Cesaro summable to the limit L in $\left(X,g\right)$ if

$\underset{n}{lim}\frac{1}{n}\sum _{k=1}^{n}\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)=0,$

and we write it as ${x}_{k}⟶L\left(|{\sigma }_{1}{|}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$. In this case L is called the $|{\sigma }_{1}{|}_{g}\left({\mathrm{\Delta }}^{m}\right)\text{-}lim$ of x.

Definition 3.4 A sequence $x=\left({x}_{k}\right)$ is said to be strongly ${\mathrm{\Delta }}^{m}$-lacunary strongly summable to the limit L in $\left(X,g\right)$ if

$\underset{r}{lim}\frac{1}{{h}_{r}}\sum _{k\in {I}_{r}}\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)=0,$

and we write it as ${x}_{k}⟶L\left({N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$. In this case L is called the ${N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\text{-}lim$ of x.

Theorem 3.1 Let θ be a lacunary sequence and $\left(X,g\right)$ be a paranormed space. Then

(i) If ${x}_{k}⟶L\left({N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$, then ${x}_{k}⟶L\left({S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$ and the inclusion is strict;

(ii) If x is a ${\mathrm{\Delta }}^{m}$-bounded sequence and ${x}_{k}⟶L\left({S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$, then ${x}_{k}⟶L\left({N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$;

(iii) ${l}_{g}^{\mathrm{\infty }}\left({\mathrm{\Delta }}^{m}\right)\cap {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)={l}_{g}^{\mathrm{\infty }}\left({\mathrm{\Delta }}^{m}\right)\cap {N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$.

Proof (i) If $\epsilon >0$ and ${x}_{k}\to L\left({N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$, we can write

$\sum _{k\in {I}_{r}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \underset{g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon }{\sum _{k\in {I}_{r}}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon |\left\{k\in {I}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon \right\}|,$

which yields the result.

In order to prove that the inclusion ${N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\subset {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$ is proper, let θ be given and $X={N}_{0}^{\theta }\left(\mathrm{\Delta },\frac{1}{{h}_{r}}\right)=\left\{x=\left({x}_{k}\right):|\frac{1}{{h}_{r}}{\sum }_{k\in {I}_{r}}\mathrm{\Delta }{x}_{k}{|}^{\frac{1}{{h}_{r}}}\to 0,r\to \mathrm{\infty }\right\}$ with the paranorm $g\left(x\right)=|{x}_{1}|+{sup}_{r}|\frac{1}{{h}_{r}}{\sum }_{k\in {I}_{r}}\mathrm{\Delta }{x}_{k}{|}^{\frac{1}{{h}_{r}}}$. Define $x=\left({x}_{k}\right)$ to be $2{h}_{r}{1}^{{h}_{r}}$ at the first term in ${I}_{r}$ for every $r\ge 1$, ${x}_{k}={h}_{r}\left({1}^{{h}_{r}}-{2}^{{h}_{r}}-\cdots -{\left(k-1\right)}^{{h}_{r}}\right)$ between the second term and $\left(\left[\sqrt{{h}_{r}}\right]+1\right)$th term in ${I}_{r}$, ${x}_{k}={h}_{r}\left({1}^{{h}_{r}}-{2}^{{h}_{r}}-\cdots -{\left(\left[\sqrt{{h}_{r}}\right]\right)}^{{h}_{r}}\right)$ at the $\left(\left[\sqrt{{h}_{r}}\right]+2\right)$th term in ${I}_{r}$ and ${x}_{k}=0$ otherwise.

We see that

and

Note that x is not Δ-bounded in $\left(X,g\right)$. We have, for every $\epsilon >0$,

$\frac{1}{{h}_{r}}|\left\{k\in {I}_{r}:g\left(\mathrm{\Delta }{x}_{k}\right)\ge \epsilon \right)\right\}|=\frac{\left[\sqrt{{h}_{r}}\right]}{{h}_{r}}\to 0$

as $r\to \mathrm{\infty }$, i.e., ${x}_{k}\to 0\left({S}_{g}^{\theta }\left(\mathrm{\Delta }\right)\right)$. On the other hand,

$\frac{1}{{h}_{r}}\sum _{k\in {I}_{r}}g\left(\mathrm{\Delta }{x}_{k}\right)=\frac{1}{{h}_{r}}\frac{\left[\sqrt{{h}_{r}}\right]\left(\left[\sqrt{{h}_{r}}\right]+1\right)}{2}\to \frac{1}{2}\ne 0;$

hence ${x}_{k}↛0\left({N}_{g}^{\theta }\left(\mathrm{\Delta }\right)\right)$.

(ii) Suppose that ${x}_{k}\to L\left({S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\right)$ and say $g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\le M$ for all k. Given $\epsilon >0$, we get

$\begin{array}{rl}\frac{1}{{h}_{r}}\sum _{k\in {I}_{r}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)& =\frac{1}{{h}_{r}}\underset{g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon }{\sum _{k\in {I}_{r}}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)+\frac{1}{{h}_{r}}\underset{g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)<\epsilon }{\sum _{k\in {I}_{r}}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\\ \le \frac{M}{{h}_{r}}|\left\{k\in {I}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\ge \epsilon \right\}|+\epsilon ,\end{array}$

from which the result follows.

(iii) This is an immediate consequence of (i) and (ii). □

Corollary 3.1 If ${x}_{k}\to L\left(|{\sigma }_{1}{|}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$, then ${x}_{k}\to L\left({S}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$. If $x\in {l}_{g}^{\mathrm{\infty }}\left({\mathrm{\Delta }}^{m}\right)$ and if ${x}_{k}\to L\left({S}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$, then ${x}_{k}\to L\left(|{\sigma }_{1}{|}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$.

Theorem 3.2 Let θ be a lacunary sequence and $\left(X,g\right)$ be a paranormed space, then ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)={S}_{g}\left({\mathrm{\Delta }}^{m}\right)$ if and only if $1<{lim inf}_{r}{q}_{r}\le {lim sup}_{r}{q}_{r}<\mathrm{\infty }$.

Proof Suppose that $liminf{q}_{r}>1$, then there exists a $\delta >0$ such that ${q}_{r}⩾1+\delta$ for sufficiently large r, which implies that

$\frac{{h}_{r}}{{k}_{r}}⩾\frac{\delta }{1+\delta }.$

If ${x}_{k}\to L\left({S}_{g}\left({\mathrm{\Delta }}^{m}\right)\right)$, then for every $\epsilon >0$ and sufficiently large r, we have

$\begin{array}{rl}\frac{1}{{k}_{r}}|\left\{k⩽{k}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)⩾\epsilon \right\}|& ⩾\frac{1}{{k}_{r}}|\left\{k\in {I}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)⩾\epsilon \right\}|\\ ⩾\frac{\delta }{1+\delta }\frac{1}{{h}_{r}}|\left\{k\in {I}_{r}:g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)⩾\epsilon \right\}|,\end{array}$

which proves the ${S}_{g}\left({\mathrm{\Delta }}^{m}\right)\subset {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$.

Conversely, suppose that $liminf{q}_{r}=1$. Since θ is lacunary, we can select a subsequence $\left({k}_{{r}_{j}}\right)$ of θ satisfying $\frac{{k}_{{r}_{j}}}{{k}_{{r}_{j}-1}}<1+\frac{1}{j}$ and $\frac{{k}_{{r}_{j}-1}}{{k}_{{r}_{\left(j-1\right)}}}>j$, where ${r}_{j}⩾{r}_{j-1}+2$ and $X={N}_{0}^{\theta }\left(\mathrm{\Delta },\frac{1}{{h}_{r}}\right)=\left\{x=\left({x}_{k}\right):|\frac{1}{{h}_{r}}{\sum }_{k\in {I}_{r}}\mathrm{\Delta }{x}_{k}{|}^{\frac{1}{{h}_{r}}}\to 0,r\to \mathrm{\infty }\right\}$ with the paranorm $g\left(x\right)=|{x}_{1}|+{sup}_{r}|\frac{1}{{h}_{r}}{\sum }_{k\in {I}_{r}}\mathrm{\Delta }{x}_{k}{|}^{\frac{1}{{h}_{r}}}$.

Now define a sequence by

$\mathrm{\Delta }{x}_{k}=\left\{\begin{array}{cc}{h}_{r}+k,\hfill & k\in {I}_{{r}_{\left(j\right)}},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$

We can see that

$g\left(\mathrm{\Delta }{x}_{k}\right)=\left\{\begin{array}{cc}1,\hfill & k\in {I}_{{r}_{\left(j\right)}},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$

and hence x is Δ-bounded in $\left(X,g\right)$.

We can see that $x\notin {N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$, but $x\in {\sigma }_{g}^{1}\left({\mathrm{\Delta }}^{m}\right)$. Theorem 3.1(ii) implies that $x\notin {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$, but it follows from Corollary 3.1 that $x\in {S}_{g}\left({\mathrm{\Delta }}^{m}\right)$. Hence ${S}_{g}\left({\mathrm{\Delta }}^{m}\right)\not\subset {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$ and ${S}_{g}\left({\mathrm{\Delta }}^{m}\right)\subset {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$ implies that $liminf{q}_{r}>1$.

To show for any lacunary sequence θ, ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\subset {S}_{g}\left({\mathrm{\Delta }}^{m}\right)$ implies $limsup{q}_{r}<\mathrm{\infty }$, the same technique of Lemma 3 of [20] can be used. Now suppose that $limsup{q}_{r}=\mathrm{\infty }$. Consider the same space defined above and the sequence defined by

$\mathrm{\Delta }{x}_{i}=\left\{\begin{array}{cc}{h}_{r}+i,\hfill & {k}_{{r}_{j}-1}

Then we get

$g\left(\mathrm{\Delta }{x}_{i}\right)=\left\{\begin{array}{cc}1,\hfill & {k}_{{r}_{j}-1}

Then $x\in {N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$, but $x\notin {\sigma }_{g}^{1}\left({\mathrm{\Delta }}^{m}\right)$. By Theorem 3.1(i) we conclude that $x\in {S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$, but by Corollary 3.1 that $x\notin {S}_{g}\left({\mathrm{\Delta }}^{m}\right)$. Hence, ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\not\subset {S}_{g}\left({\mathrm{\Delta }}^{m}\right)$. This completes the proof. □

Definition 3.5 Let f be a modulus function. Then a sequence $x=\left({x}_{k}\right)$ is lacunary strongly p-Cesaro summable to L with respect to f in $\left(X,g\right)$ if

$\underset{r}{lim}\frac{1}{{h}_{r}}\sum _{kϵ{I}_{r}}{\left[f\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)\right]}^{{p}_{k}}=0.$

In this case, we write ${x}_{k}\to L\left({N}_{g}^{\theta }\left(f,{\mathrm{\Delta }}^{m},p\right)\right)$. If we take ${p}_{k}=1$ for all $k\in \mathbb{N}$, we say ${x}_{k}\to L\left({N}_{g}^{\theta }\left(f,{\mathrm{\Delta }}^{m}\right)\right)$.

Lemma 3.1 Let f be a modulus function and let $0<\delta <1$. Then for each $x>\delta$ we have $f\left(x\right)⩽2f\left(1\right){\delta }^{-1}x$ [21].

Theorem 3.3 Let f be a modulus function and $\left(X,g\right)$ be a paranormed space. Then ${N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)\subset {N}_{g}^{\theta }\left(f,{\mathrm{\Delta }}^{m}\right)$.

Proof Let $x\in {N}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)$. Then we have ${\tau }_{r}=\frac{1}{{h}_{r}}{\sum }_{k\in {I}_{r}}g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\to 0$ as $r\to \mathrm{\infty }$ for some L.

Let $\epsilon >0$ and choose δ with $0<\delta <1$ such that $f\left(u\right)<\epsilon$ for u with $0\le u\le \delta$. Then we can write

$\begin{array}{rl}\frac{1}{{h}_{r}}\sum _{k\in {I}_{r}}f\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)=& \frac{1}{{h}_{r}}\underset{g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\le \delta }{\sum _{k\in {I}_{r}}}f\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)\\ +\frac{1}{{h}_{r}}\underset{g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)>\delta }{\sum _{k\in {I}_{r}}}f\left(g\left({\mathrm{\Delta }}^{m}{x}_{k}-L\right)\right)\\ \le & \frac{1}{{h}_{r}}\left({h}_{r}\delta \right)+\frac{1}{{h}_{r}}2f\left(1\right){\delta }^{-1}{h}_{r}{\tau }_{r}\end{array}$

from Lemma 3.1. Therefore $x\in {N}_{g}^{\theta }\left(f,{\mathrm{\Delta }}^{m}\right)$. □

Theorem 3.4 Let $0<{inf}_{k}{p}_{k}\le {p}_{k}\le {sup}_{k}{p}_{k}<\mathrm{\infty }$. Then ${S}_{g}^{\theta }\left({\mathrm{\Delta }}^{m}\right)={N}_{g}^{\theta }\left(f,{\mathrm{\Delta }}^{m},p\right)$ if and only if f is bounded.

Proof Following the technique applied for establishing Theorem 3.16 of [22], we can prove the theorem. □

## References

1. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.

2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73–74.

3. Fridy JA: Lacunary statistical summability. J. Math. Anal. Appl. 1993, 173: 497–504. 10.1006/jmaa.1993.1082

4. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233(2):142–149. 10.1016/j.cam.2009.07.005

5. Kolk E: The statistical convergence in Banach spaces. Tartu ülik. Toim. 1991, 928: 41–52.

6. Maddox IJ: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc. 1988, 104: 141–145. 10.1017/S0305004100065312

7. Çakallı H: On statistical convergence in topological groups. Pure Appl. Math. Sci. 1996, 43: 27–31.

8. Karakuş S: Statistical convergence on probabilistic normed spaces. Math. Commun. 2007, 12: 11–23.

9. Alotaibi A, Alroqi A: Statistical convergence in a paranormed space. J. Inequal. Appl. 2012. doi:10.1186/1029–242X-2012–39

10. Freedman AR, Sember JJ: Densities and summability. Pac. J. Math. 1981, 95: 293–305. 10.2140/pjm.1981.95.293

11. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

12. Connor JS: The statistical and strong p -Cesaro convergence of sequences. Analysis 1988, 8: 47–63.

13. Kızmaz H: On certain sequence spaces. Can. Math. Bull. 1981, 24(2):169–176. 10.4153/CMB-1981-027-5

14. Et M, Çolak R: On some generalized difference sequence spaces. Soochow J. Math. 1995, 21(4):377–386.

15. Et M, Nuray F:${\mathrm{\Delta }}^{m}$-statistical convergence. Indian J. Pure Appl. Math. 2001, 32(6):961–969.

16. Et M: Spaces of Cesaro difference sequences of order r defined by a modulus function in a locally convex space. Taiwan. J. Math. 2006, 10(4):865–879.

17. Et M, Choudhary B, Tripathy BC: On some classes of sequences defined by sequences of Orlicz functions. Math. Inequal. Appl. 2006, 9(2):335–342.

18. Mursaleen M: Elements of Metric Spaces. Anamaya Publishers, New Delhi; 2005.

19. Ruckle WH: FK spaces in which the sequence of coordinate vectors in bounded. Can. J. Math. 1973, 25(5):973–975.

20. Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160(1):43–51. 10.2140/pjm.1993.160.43

21. Pehlivan S, Fisher B: Some sequence spaces defined by a modulus function. Math. Slovaca 1995, 45(3):275–280.

22. Tripathy BC, Et M: On generalized difference lacunary statistical convergence. Stud. Univ. Babeş-Bolyai, Math. 2005, 50(1):119–130.

## Acknowledgements

The author would like to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

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Correspondence to Selma Altundağ.

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Altundağ, S. On generalized difference lacunary statistical convergence in a paranormed space. J Inequal Appl 2013, 256 (2013). https://doi.org/10.1186/1029-242X-2013-256