# On a new application of almost increasing sequences

## Abstract

In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with ${|\overline{N},{p}_{n}|}_{k}$ summability factors of an infinite series. In the present paper, we have generalized this theorem on the $\phi -{|A,{p}_{n}|}_{k}$ summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.

MSC:40D15, 40F05, 40G99.

## 1 Introduction

Let $\sum {a}_{n}$ be a given infinite series with the partial sums $\left({s}_{n}\right)$. We denote by ${t}_{n}$ the n th $\left(C,1\right)$ mean of the sequence $\left({s}_{n}\right)$. The series $\sum {a}_{n}$ is said to be summable ${|C,1|}_{k}$, $k\ge 1$, if (see )

$\sum _{n=1}^{\mathrm{\infty }}{n}^{k-1}{|{t}_{n}-{t}_{n-1}|}^{k}<\mathrm{\infty }.$
(1)

Let $\left({p}_{n}\right)$ be a sequence of positive numbers such that

(2)

The sequence-to-sequence transformation

${\sigma }_{n}=\frac{1}{{P}_{n}}\sum _{v=0}^{n}{p}_{v}{s}_{v}$
(3)

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

$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{{P}_{n}}{{p}_{n}}\right)}^{k-1}{|{\sigma }_{n}-{\sigma }_{n-1}|}^{k}<\mathrm{\infty }.$
(4)

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

${A}_{n}\left(s\right)=\sum _{v=0}^{n}{a}_{nv}{s}_{v},\phantom{\rule{1em}{0ex}}n=0,1,\dots .$
(5)

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

$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{{P}_{n}}{{p}_{n}}\right)}^{k-1}{|\overline{\mathrm{\Delta }}{A}_{n}\left(s\right)|}^{k}<\mathrm{\infty },$
(6)

where

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

Let $\left({\phi }_{n}\right)$ be any sequence of positive real numbers. The series $\sum {a}_{n}$ is said to be summable $\phi -|A,{p}_{n}{|}_{k}$, $k\ge 1$, if (see )

$\sum _{n=1}^{\mathrm{\infty }}{\phi }_{n}^{k-1}|\overline{\mathrm{\Delta }}{A}_{n}\left(s\right){|}^{k}<\mathrm{\infty }.$
(7)

If we take ${\phi }_{n}=\frac{{P}_{n}}{{p}_{n}}$, then $\phi -|A,{p}_{n}{|}_{k}$ summability reduces to ${|A,{p}_{n}|}_{k}$ summability. Also, if we take ${\phi }_{n}=\frac{{P}_{n}}{{p}_{n}}$ and ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$, then we get $|\overline{N},{p}_{n}{|}_{k}$ summability. Furthermore, if we take ${\phi }_{n}=n$, ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$ and ${p}_{n}=1$ for all values of n, $\phi -|A,{p}_{n}{|}_{k}$ reduces to ${|C,1|}_{k}$ summability. Finally, if we take ${\phi }_{n}=n$ and ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$, then we get ${|R,{p}_{n}|}_{k}$ summability (see ).

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:

${\overline{a}}_{nv}=\sum _{i=v}^{n}{a}_{ni},\phantom{\rule{1em}{0ex}}n,v=0,1,\dots$
(8)

and

${\stackrel{ˆ}{a}}_{00}={\overline{a}}_{00}={a}_{00},\phantom{\rule{2em}{0ex}}{\stackrel{ˆ}{a}}_{nv}={\overline{a}}_{nv}-{\overline{a}}_{n-1,v},\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(9)

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

${A}_{n}\left(s\right)=\sum _{v=0}^{n}{a}_{nv}{s}_{v}=\sum _{v=0}^{n}{\overline{a}}_{nv}{a}_{v}$
(10)

and

$\overline{\mathrm{\Delta }}{A}_{n}\left(s\right)=\sum _{v=0}^{n}{\stackrel{ˆ}{a}}_{nv}{a}_{v}.$
(11)

## 2 Known result

Many works have been done dealing with $|\overline{N},{p}_{n}{|}_{k}$ summability factors of infinite series (see ). Among them, in , the following main theorem has been proved.

Theorem A Let $\left({X}_{n}\right)$ be a positive non-decreasing sequence and let there be sequences $\left({\beta }_{n}\right)$ and $\left({\lambda }_{n}\right)$ such that (12) (13) (14) (15)

are satisfied. Furthermore, if $\left({p}_{n}\right)$ is a sequence of positive numbers such that (16) (17)

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

## 3 The main result

The aim of this paper is to generalize Theorem A for $\phi -{|A,{p}_{n}|}_{k}$ summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence $\left({c}_{n}\right)$ is said to be almost increasing if there exists a positive increasing sequence $\left({b}_{n}\right)$ and two positive constants A and B such that $A{b}_{n}\le {c}_{n}\le B{b}_{n}$ (see ). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example ${b}_{n}=n{e}^{{\left(-1\right)}^{n}}$. Also, one can find some results dealing with absolute almost convergent sequences (see ). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.

Theorem Let $A=\left({a}_{nv}\right)$ be a positive normal matrix such that (18) (19) (20) (21)

Let $\left({X}_{n}\right)$ be an almost increasing sequence and $\left(\frac{{\phi }_{n}{p}_{n}}{{P}_{n}}\right)$ be a non-increasing sequence. If conditions (12)-(16) and

$\sum _{n=1}^{m}{\phi }_{n}^{k-1}{\left(\frac{{p}_{n}}{{P}_{n}}\right)}^{k}{|{s}_{n}|}^{k}=O\left({X}_{m}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}m\to \mathrm{\infty },$
(22)

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

Remark It should be noted that if we take $\left({X}_{n}\right)$ as a positive non-decreasing sequence, ${\phi }_{n}=\frac{{P}_{n}}{{p}_{n}}$ and ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$, then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘$\left(\frac{{\phi }_{n}{p}_{n}}{{P}_{n}}\right)$ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.

Lemma 

Under the conditions on $\left({X}_{n}\right)$, $\left({\beta }_{n}\right)$ and $\left({\lambda }_{n}\right)$ as taken in the statement of the theorem, we have the following: (23) (24)

Proof of the Theorem Let $\left({T}_{n}\right)$ denote A-transform of the series $\sum {a}_{n}{\lambda }_{n}$. Then we have, by (10) and (11),

$\overline{\mathrm{\Delta }}{T}_{n}=\sum _{v=1}^{n}{\stackrel{ˆ}{a}}_{nv}{\lambda }_{v}{a}_{v}.$

Applying Abel’s transformation to this sum, we get that

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

To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that

Now, when $k>1$, applying Hölder’s inequality with indices k and $\stackrel{´}{k}$, where $1/k+1/\stackrel{´}{k}=1$, we have that

by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that $v{\beta }_{v}=O\left(\frac{1}{{X}_{v}}\right)=O\left(1\right)$ by (23), we get that

by virtue of the hypotheses of the theorem and the lemma. Finally, as in ${T}_{n}\left(1\right)$, we have that

This completes the proof of the theorem. If we take ${\phi }_{n}=\frac{{P}_{n}}{{p}_{n}}$, then we get a result concerning the ${|A,{p}_{n}|}_{k}$ summability factors. If we take ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$, then we have another result dealing with ${|\overline{N},{p}_{n},{\phi }_{n}|}_{k}$ summability. If we take ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$ and ${p}_{n}=1$ for all values of n, then we get a result dealing with ${|C,1,{\phi }_{n}|}_{k}$ summability. If we take ${\phi }_{n}=n$, ${a}_{nv}=\frac{{p}_{v}}{{P}_{n}}$ and ${p}_{n}=1$ for all values of n, then we get a result for ${|C,1|}_{k}$ summability. □

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Özarslan, H., Keten, A. On a new application of almost increasing sequences. J Inequal Appl 2013, 13 (2013). https://doi.org/10.1186/1029-242X-2013-13 