# A new note on absolute matrix summability

## Abstract

In the present paper, we have proved theorems dealing with matrix summability factors by using quasi β-power increasing sequences. Some new results have also been obtained.

MSC:40D15, 40F05, 40G99.

## 1 Introduction

A positive sequence $\left({\gamma }_{n}\right)$ is said to be quasi β-power increasing sequence if there exists a constant $K=K\left(\beta ,\gamma \right)\ge 1$ such that $K{n}^{\beta }{\gamma }_{n}\ge {m}^{\beta }{\gamma }_{m}$ holds for all $n\ge m\ge 1$ . A sequence $\left({\lambda }_{n}\right)$ is said to be of bounded variation, denote by $\left({\lambda }_{n}\right)\in \mathcal{BV}$, if ${\sum }_{n=1}^{\mathrm{\infty }}|\mathrm{\Delta }{\lambda }_{n}|={\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n}-{\lambda }_{n+1}|<\mathrm{\infty }$. Let $\sum {a}_{n}$ be a given infinite series with the partial sums $\left({s}_{n}\right)$. Let $\left({p}_{n}\right)$ be a sequence of positive numbers such that

(1)

The sequence-to-sequence transformation

${\sigma }_{n}=\frac{1}{{P}_{n}}\sum _{v=0}^{n}{p}_{v}{s}_{v}$
(2)

defines the sequence $\left({\sigma }_{n}\right)$ of the $\left(\overline{N},{p}_{n}\right)$ mean of the sequence $\left({s}_{n}\right)$, generated by the sequence of coefficients $\left({p}_{n}\right)$ .

The series $\sum {a}_{n}$ is said to be summable $|\overline{N},{p}_{n}{|}_{k}$, $k\ge 1$ if 

$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{{P}_{n}}{{p}_{n}}\right)}^{k-1}{|{\sigma }_{n}-{\sigma }_{n-1}|}^{k}<\mathrm{\infty }.$
(3)

Let $A=\left({a}_{nv}\right)$ be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence $s=\left({s}_{n}\right)$ to $As=\left({A}_{n}\left(s\right)\right)$, where

${A}_{n}\left(s\right)=\sum _{v=0}^{n}{a}_{nv}{s}_{v},\phantom{\rule{1em}{0ex}}n=0,1,\dots .$
(4)

The series $\sum {a}_{n}$ is said to be summable $|A,{p}_{n}{|}_{k}$, $k\ge 1$ if 

$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{{P}_{n}}{{p}_{n}}\right)}^{k-1}{|\overline{\mathrm{\Delta }}{A}_{n}\left(s\right)|}^{k}<\mathrm{\infty },$
(5)

where

$\overline{\mathrm{\Delta }}{A}_{n}\left(s\right)={A}_{n}\left(s\right)-{A}_{n-1}\left(s\right).$

Before stating the main theorem, we must first introduce some further notations.

Given a normal matrix $A=\left({a}_{nv}\right)$, we associate two lower semimatrices $\overline{A}=\left({\overline{a}}_{nv}\right)$ and $\stackrel{ˆ}{A}=\left({\stackrel{ˆ}{a}}_{nv}\right)$ as follows:

${\overline{a}}_{nv}=\sum _{i=v}^{n}{a}_{ni},\phantom{\rule{1em}{0ex}}n,v=0,1,\dots$
(6)

and

${\stackrel{ˆ}{a}}_{00}={\overline{a}}_{00}={a}_{00},\phantom{\rule{2em}{0ex}}{\stackrel{ˆ}{a}}_{nv}={\overline{a}}_{nv}-{\overline{a}}_{n-1,v},\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(7)

It may be noted that $\overline{A}$ and $\stackrel{ˆ}{A}$ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have

${A}_{n}\left(s\right)=\sum _{v=0}^{n}{a}_{nv}{s}_{v}=\sum _{v=0}^{n}{\overline{a}}_{nv}{a}_{v}$
(8)

and

$\overline{\mathrm{\Delta }}{A}_{n}\left(s\right)=\sum _{v=0}^{n}{\stackrel{ˆ}{a}}_{nv}{a}_{v}.$
(9)

## 2 Known result

Recently, many authors have come up with theorems dealing with the applications of power increasing sequences [1, 57]. Among them, Bor and Özarslan have proved two theorems for $|\overline{N},{p}_{n}{|}_{k}$ summability method by using quasi β-power increasing sequence . Their theorems are as follows.

Theorem A Let $\left({X}_{n}\right)$ be a quasi β-power increasing sequence for some $0<\beta <1$, and let there be sequences $\left({\beta }_{n}\right)$ and $\left({\lambda }_{n}\right)$ such that

$|\mathrm{\Delta }{\lambda }_{n}|\le {\beta }_{n},$
(10)
(11)
$\sum _{n=1}^{\mathrm{\infty }}n|\mathrm{\Delta }{\beta }_{n}|{X}_{n}<\mathrm{\infty },$
(12)
(13)

If

$\sum _{v=1}^{n}\frac{{|{s}_{v}|}^{k}}{v}=O\left({X}_{n}\right),$
(14)
$\sum _{n=1}^{m}\frac{{p}_{n}}{{P}_{n}}{|{s}_{n}|}^{k}=O\left({X}_{m}\right),\phantom{\rule{1em}{0ex}}m\to \mathrm{\infty },$
(15)

then $\sum {a}_{n}{\lambda }_{n}$ is summable $|\overline{N},{p}_{n}{|}_{k}$, $k\ge 1$.

Theorem B Let $\left({X}_{n}\right)$ be a quasi β-power increasing sequence for some $0<\beta <1$, and let sequences $\left({\beta }_{n}\right)$ and $\left({\lambda }_{n}\right)$ satisfy conditions (10)-(13) and (15). If

$\sum _{n=1}^{\mathrm{\infty }}{P}_{n}|\mathrm{\Delta }{\beta }_{n}|{X}_{n}<\mathrm{\infty },$
(16)
$\sum _{n=1}^{m}\frac{{|{s}_{n}|}^{k}}{{P}_{n}}=O\left({X}_{m}\right),$
(17)

then $\sum {a}_{n}{\lambda }_{n}$ is summable $|\overline{N},{p}_{n}{|}_{k}$, $k\ge 1$.

## 3 The main result

The aim of this paper is to generalize Theorem A and Theorem B to $|A,{p}_{n}{|}_{k}$ summability. Now, we shall prove the following two theorems.

Theorem 1 Let $A=\left({a}_{nv}\right)$ be a positive normal matrix such that

${\overline{a}}_{n0}=1,\phantom{\rule{1em}{0ex}}n=0,1,\dots ,$
(18)
(19)
${a}_{nn}=O\left(\frac{{p}_{n}}{{P}_{n}}\right),$
(20)

and $\left({X}_{n}\right)$ is a quasi β-power increasing sequence for some $0<\beta <1$. If all the conditions of Theorem A and

$\left({\lambda }_{n}\right)\in \mathcal{BV}$
(21)

are satisfied, then the series $\sum {a}_{n}{\lambda }_{n}$ is summable ${|A,{p}_{n}|}_{k}$, $k\ge 1$.

In the special case of ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$, this theorem reduces to Theorem A.

Theorem 2 Let $A=\left({a}_{nv}\right)$ be a positive normal matrix as in Theorem 1, and let $\left({X}_{n}\right)$ is a quasi β-power increasing sequence for some $0<\beta <1$. If all the conditions of Theorem B and (21) are satisfied, then the series $\sum {a}_{n}{\lambda }_{n}$ is summable ${|A,{p}_{n}|}_{k}$, $k\ge 1$.

We need following lemmas for the proof of our theorems.

Lemma 1 

Let $\left({X}_{n}\right)$ be a quasi β-power increasing sequence for some $0<\beta <1$. If conditions (11) and (12) satisfied, then

(22)
$\sum _{n=1}^{\mathrm{\infty }}{X}_{n}{\beta }_{n}<\mathrm{\infty }.$
(23)

Lemma 2 Let $\left({X}_{n}\right)$ be a quasi β-power increasing sequence for some $0<\beta <1$. If conditions (11) and (16) are satisfied, then

${P}_{n}{\beta }_{n}{X}_{n}=O\left(1\right),$
(24)
$\sum _{n=1}^{\mathrm{\infty }}{p}_{n}{\beta }_{n}{X}_{n}<\mathrm{\infty }.$
(25)

The proof of Lemma 2 is similar to that of Bor in  and hence is omitted.

## 4 Proof of Theorem 1

Let $\left({T}_{n}\right)$ denote A-transform of the series $\sum {a}_{n}{\lambda }_{n}$. Then by (8), (9) and applying Abel’s transformation, we have

$\begin{array}{rcl}\overline{\mathrm{\Delta }}{T}_{n}& =& \sum _{v=1}^{n}{\stackrel{ˆ}{a}}_{nv}{a}_{v}{\lambda }_{v}\\ =& \sum _{v=1}^{n-1}{\mathrm{\Delta }}_{v}\left({\stackrel{ˆ}{a}}_{nv}{\lambda }_{v}\right)\sum _{k=1}^{v}{a}_{k}+{\stackrel{ˆ}{a}}_{nn}{\lambda }_{n}\sum _{v=1}^{n}{a}_{v}\\ =& \sum _{v=1}^{n-1}\left({\stackrel{ˆ}{a}}_{nv}{\lambda }_{v}-{\stackrel{ˆ}{a}}_{n,v+1}{\lambda }_{v+1}\right){s}_{v}+{a}_{nn}{\lambda }_{n}{s}_{n}\\ =& \sum _{v=1}^{n-1}\left({\stackrel{ˆ}{a}}_{nv}{\lambda }_{v}-{\stackrel{ˆ}{a}}_{n,v+1}{\lambda }_{v+1}-{\stackrel{ˆ}{a}}_{n,v+1}{\lambda }_{v}+{\stackrel{ˆ}{a}}_{n,v+1}{\lambda }_{v}\right){s}_{v}+{a}_{nn}{\lambda }_{n}{s}_{n}\\ =& \sum _{v=1}^{n-1}{\mathrm{\Delta }}_{v}\left({\stackrel{ˆ}{a}}_{nv}\right){\lambda }_{v}{s}_{v}+\sum _{v=1}^{n-1}{\stackrel{ˆ}{a}}_{n,v+1}\mathrm{\Delta }{\lambda }_{v}{s}_{v}+{a}_{nn}{\lambda }_{n}{s}_{n}\\ =& {T}_{n,1}+{T}_{n,2}+{T}_{n,3}\phantom{\rule{1em}{0ex}}\text{say}.\end{array}$

Since

$|{T}_{n,1}+{T}_{n,2}+{T}_{n,3}{|}^{k}\le {3}^{k}\left({|{T}_{n,1}|}^{k}+{T}_{n,2}{|}^{k}+{T}_{n,3}{|}^{k}\right),$

to complete the proof of the Theorem 1, it is sufficient to show that

(26)

First, applying Hölder’s inequality with indices k and ${k}^{\prime }$, where $k>1$ and $\frac{1}{k}+\frac{1}{{k}^{\prime }}=1$, we get that

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Since $\left({\lambda }_{n}\right)\in \mathcal{BV}$ by (21), applying Hölder’s inequality with the same indices as those above, we have

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Finally, by following the similar process as in ${T}_{n,1}$, we have that

So, we get

This completes the proof of Theorem 1.

## 5 Proof of Theorem 2

Using Lemma 2 and proceeding as in the proof of Theorem 1, replacing ${\sum }_{v=1}^{m}{\beta }_{v}{|{s}_{v}|}^{k}$ by ${\sum }_{v=1}^{m}{\beta }_{v}{P}_{v}\left(\frac{{|{s}_{v}|}^{k}}{{P}_{v}}\right)$, we can easily prove Theorem 2.

If we take ${p}_{n}=1$ in these theorems, then we have two new results dealing with ${|A|}_{k}$ summability factors of infinite series. Also, if we take $k=1$, then we obtain another two new results concerning $|A|$ summability. Finally, by taking $\left({X}_{n}\right)$ as almost increasing sequence in the theorems, we get new results dealing with $|A,{p}_{n}{|}_{k}$ summability factors of infinite series.

## References

1. Leindler L: A new application of quasi power increasing sequences. Publ. Math. (Debr.) 2001, 58: 791–796.

2. Hardy GH: Divergent Series. Oxford University Press, Oxford; 1949.

3. Bor H: On two summability methods. Math. Proc. Camb. Philos. Soc. 1985, 97(1):147–149. 10.1017/S030500410006268X

4. Sulaiman WT: Inclusion theorems for absolute matrix summability methods of an infinite series (IV). Indian J. Pure Appl. Math. 2003, 34(11):1547–1557.

5. Bor H, Özarslan HS: On the quasi power increasing sequences. J. Math. Anal. Appl. 2002, 276: 924–929. 10.1016/S0022-247X(02)00494-8

6. Bor H, Debnath L: Quasi β -power increasing sequences. Int. J. Math. Math. Sci. 2004, 44: 2371–2376.

7. Bor H: A general note on increasing sequences. J. Inequal. Pure Appl. Math. 2007, 8/3: 1–7.

8. Bor H: On absolute summability factors for ${|\overline{N},{p}_{n}|}_{k}$ summability. Comment. Math. Univ. Carol. 1991, 32/3: 435–439.

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Correspondence to Hikmet S Özarslan.

### Competing interests

The authors declare that they have no competing interests.

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

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Özarslan, H.S., Yavuz, E. A new note on absolute matrix summability. J Inequal Appl 2013, 474 (2013). https://doi.org/10.1186/1029-242X-2013-474 