# Correction to: On statistical convergence and strong Cesàro convergence by moduli

The Original Article was published on 14 November 2019

## Abstract

We correct a logic mistake in our paper “On statistical convergence and strong Cesàro convergence by moduli” (León-Saavedra et al. in J. Inequal. Appl. 23:298, 2019).

## 1 Introduction

It has come to our attention that there is a logic mistake with the converse of some results in our paper [1]. These converse of these results are not central in the papers, but they could be interested in its own right. The next result correct Proposition 2.9 in [1].

### Proposition 1.1

1. (a)

If all statistical convergent sequences are f-statistical convergent then f is a compatible modulus function.

2. (b)

If all strong Cesàro convergent sequences are f-strong Cesàro convergent then f is a compatible modulus function.

### Proof

Let $$\varepsilon _{n}$$ be a decreasing sequence converging to 0. Since f is not compatible, there exists $$c>0$$ such that, for each k, there exists $$m_{k}$$ such that $$f(m_{k}\varepsilon _{k})>cf(m_{k})$$. Moreover, we can select $$m_{k}$$ inductively satisfying

\begin{aligned} 1-\varepsilon _{k+1}-\frac{1}{m_{k+1}}> \frac{(1-\varepsilon _{k})m_{k}}{m_{k+1}}. \end{aligned}
(1.1)

Now we use an standard argument used to construct subsets with prescribed densities. Let us denote $$\lfloor x\rfloor$$ the integer part of $$x\in \mathbb{R}$$. Set $$n_{k}=\lfloor m_{k}\varepsilon _{k}\rfloor +1$$. And extracting a subsequence if it is necessary, we can assume that $$n_{1}< n_{2}<\cdots$$ , $$m_{1}< m_{2}<\cdots$$ . Thus, set $$A_{k}=[m_{k+1}-(n_{k+1}-n_{k})]\cap \mathbb{N}$$. Condition (1.1) guarantee that $$A_{k}\subset [m_{k},m_{k+1}]$$.

Let us denote $$A=\bigcup_{k}A_{k}$$, and $$x_{n}=\chi _{A}(n)$$. Let us prove that $$x_{n}$$ is statistical convergent to 0, but not f-statistical convergent, a contradiction. Indeed, for any m, there exists k such that $$m_{k}< m\leq m_{k+1}$$. Moreover, we can suppose without loss that $$m\in A$$, that is, $$m_{k+1}-n_{k+1}+n_{k}\leq m$$. Thus for any $$\varepsilon >0$$:

\begin{aligned} \frac{\#\{l\leq m: \vert x_{l} \vert >\varepsilon \}}{m}&\leq \frac{\#\{l\leq m_{k}: \vert x_{k} \vert >\varepsilon \}}{m_{k}}+ \frac{n_{k+1}-n_{k}}{m_{k+1}-n_{k+1}+n_{k}} \\ &\leq \frac{n_{k}}{m_{k}}+\frac{1}{\frac{m_{k+1}}{n_{k+1}-n_{k}}-1} \to 0 \end{aligned}

as $$k\to \infty$$. On the other hand, since $$\varepsilon _{k+1}<\frac{n_{k+1}}{m_{k+1}}$$

\begin{aligned} \frac{f (\#\{n< m_{k+1}: \vert x_{n} \vert >1/2\} )}{f(m_{k+1})} = \frac{f(n_{k+1})}{f(m_{k+1})} \geq \frac{f(m_{k+1}\varepsilon _{k+1})}{f(m_{k+1})}>c, \end{aligned}

which yields (a) as promised. The part (b) is same proof. Indeed, for the sequence $$(x_{n})$$ defined in part (a), we have that $$\frac{f(\sum_{k=1}^{n}|x_{n}|)}{f(n)}= \frac{f(\{k\leq n |x_{k}|>\varepsilon \})}{f(n)}$$. □

The following result corrects the converse of Theorem 3.4 in [1].

### Proposition 1.2

If all f-strong Cesàro convergent sequences are f-statistically and uniformly bounded then f must be compatible.

### Proof

Assume that f is not compatible. Thus, as in the proof in Proposition 1.1 we can construct sequences $$(\varepsilon _{k})$$, $$(m_{k})$$ such that $$f(m_{k}\varepsilon _{k})\geq c f(m_{k})$$ for some $$c>0$$. Moreover, we can construct $$(m_{k})$$ inductively, such that the sequence

\begin{aligned} r_{k}= \frac{m_{k+1}\varepsilon _{k+1}-m_{k}\varepsilon _{k}}{m_{k+1}-m_{k}} \end{aligned}

is decreasing and converging to 0. Let us consider $$x_{n}=\sum_{k=0}^{\infty}r_{k+1}\chi _{(m_{k},m_{k+1}]}(n)$$. Since $$(x_{n})$$ is decreasing, $$(x_{n})$$ if f-statistically convergent to 0. On the other hand $$f(\sum_{l=1}^{m_{k}} |x_{l}|)=f(m_{k}\varepsilon _{k})\geq cf(m_{k})$$, which gives that $$(x_{n})$$ is not f-strong Cesàro convergent, as we desired. □

The corrections have been indicated in this article and the original article [1] has been corrected.

## References

1. León-Saavedra, F., Listán-García, M.C., Pérez Fernández, F.J., Romero de la Rosa, M.P.: On statistical convergence and strong Cesàro convergence by moduli. J. Inequal. Appl. 12, Article ID 298 (2019)

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Correspondence to M. del Carmen Listán-García.

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