# Almost increasing sequences and their new applications

## Abstract

In this paper, we generalize a known theorem dealing with $|C,1{|}_{k}$ summability factors to the ${|C,\alpha |}_{k}$ summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.

MSC:26D15, 40D15, 40F05, 40G05.

## 1 Introduction

A positive sequence $\left({b}_{n}\right)$ is said to be an almost increasing sequence if there exists a positive increasing sequence $\left({c}_{n}\right)$ and two positive constants A and B such that $A{c}_{n}\le {b}_{n}\le B{c}_{n}$ (see ). Let $\sum {a}_{n}$ be a given infinite series with the sequence of partial sums $\left({s}_{n}\right)$. By ${t}_{n}^{\alpha }$ we denote the n th Cesàro mean of order α, with $\alpha >-1$, of the sequence $\left(n{a}_{n}\right)$, that is,

${t}_{n}^{\alpha }=\frac{1}{{A}_{n}^{\alpha }}\sum _{v=0}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v},$
(1)

where

(2)

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

$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}|{t}_{n}^{\alpha }{|}^{k}<\mathrm{\infty }.$
(3)

If we take $\alpha =1$, then ${|C,\alpha |}_{k}$ summability reduces to $|C,1{|}_{k}$ summability.

## 2 Known result

Many works dealing with an application of almost increasing sequences to the absolute Cesàro summability factors of infinite series have been done (see ). Among them, in , the following main theorem dealing with $|C,1{|}_{k}$ summability factors has been proved.

Theorem A Let $\left({\phi }_{n}\right)$ be a positive sequence and $\left({X}_{n}\right)$ be an almost increasing sequence. If the conditions

$\sum _{n=1}^{\mathrm{\infty }}n|{\mathrm{\Delta }}^{2}{\lambda }_{n}|{X}_{n}<\mathrm{\infty },$
(4)
$|{\lambda }_{n}|{X}_{n}=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty },$
(5)
${\phi }_{n}=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty },$
(6)
$n\mathrm{\Delta }{\phi }_{n}=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty },$
(7)
$\sum _{v=1}^{n}\frac{|{t}_{v}{|}^{k}}{v{X}_{v}^{k-1}}=O\left({X}_{n}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty }$
(8)

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

## 3 The main result

The aim of this paper is to generalize Theorem A to the ${|C,\alpha |}_{k}$ summability in the following form.

Theorem Let $\left({\phi }_{n}\right)$ be a positive sequence and let $\left({X}_{n}\right)$ be an almost increasing sequence.

If the conditions (4), (5), (6) and (7) are satisfied, and the sequence $\left({w}_{n}^{\alpha }\right)$ defined by (see )

${w}_{n}^{\alpha }=\left\{\begin{array}{cc}|{t}_{n}^{\alpha }|,\hfill & \alpha =1,\hfill \\ {max}_{1\le v\le n}|{t}_{v}^{\alpha }|,\hfill & 0<\alpha <1,\hfill \end{array}$
(9)

satisfies the condition

$\sum _{v=1}^{n}\frac{{\left({w}_{v}^{\alpha }\right)}^{k}}{v{X}_{v}^{k-1}}=O\left({X}_{n}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty },$
(10)

then the series $\sum {a}_{n}{\lambda }_{n}{\phi }_{n}$ is summable ${|C,\alpha |}_{k}$, $0<\alpha \le 1$, $\left(\alpha -1\right)k>-1$ and $k\ge 1$.

Remark It should be noted that if we take $\alpha =1$, then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘$\left(\alpha -1\right)k>-1$’ is trivial.

We need the following lemmas for the proof of our theorem.

Lemma 1 

If $0<\alpha \le 1$ and $1\le v\le n$, then

$|\sum _{p=0}^{v}{A}_{n-p}^{\alpha -1}{a}_{p}|\le \underset{1\le m\le v}{max}|\sum _{p=0}^{m}{A}_{m-p}^{\alpha -1}{a}_{p}|.$
(11)

Lemma 2 

Under the conditions (4) and (5), we have

$n{X}_{n}|\mathrm{\Delta }{\lambda }_{n}|=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}n\to \mathrm{\infty },$
(12)
$\sum _{n=1}^{\mathrm{\infty }}{X}_{n}|\mathrm{\Delta }{\lambda }_{n}|<\mathrm{\infty }.$
(13)

## 4 Proof of the Theorem

Let $\left({T}_{n}^{\alpha }\right)$ be the n th $\left(C,\alpha \right)$ mean, with $0<\alpha \le 1$, of the sequence $\left(n{a}_{n}{\lambda }_{n}{\phi }_{n}\right)$.

Then, by (1), we find that

${T}_{n}^{\alpha }=\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v}{\lambda }_{v}{\phi }_{n}.$
(14)

Thus, applying Abel’s transformation first and then using Lemma 1, we have that

$\begin{array}{c}\begin{array}{rl}{T}_{n}^{\alpha }& =\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}\mathrm{\Delta }\left({\lambda }_{v}{\phi }_{n}\right)\sum _{p=1}^{v}{A}_{n-p}^{\alpha -1}p{a}_{p}+\frac{{\lambda }_{n}{\phi }_{n}}{{A}_{n}^{\alpha }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v}\\ =\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}\left({\lambda }_{v}\mathrm{\Delta }{\phi }_{v}+{\phi }_{v+1}\mathrm{\Delta }{\lambda }_{v}\right)\sum _{p=1}^{v}{A}_{n-p}^{\alpha -1}p{a}_{p}+\frac{{\lambda }_{n}{\phi }_{n}}{{A}_{n}^{\alpha }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v},\end{array}\hfill \\ \begin{array}{rl}|{T}_{n}^{\alpha }|\le & \frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}|{\lambda }_{v}\mathrm{\Delta }{\phi }_{v}||\sum _{p=1}^{v}{A}_{n-p}^{\alpha -1}p{a}_{p}|+\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}|{\phi }_{v+1}\mathrm{\Delta }{\lambda }_{v}||\sum _{p=1}^{v}{A}_{n-p}^{\alpha -1}p{a}_{p}|\\ +\frac{|{\lambda }_{n}{\phi }_{n}|}{{A}_{n}^{\alpha }}|\sum _{v=1}^{v}{A}_{n-v}^{\alpha -1}v{a}_{v}|\\ \le & \frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}{A}_{v}^{\alpha }{w}_{v}^{\alpha }|{\lambda }_{v}||\mathrm{\Delta }{\phi }_{v}|+\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n-1}{A}_{v}^{\alpha }{w}_{v}^{\alpha }|{\phi }_{v+1}||\mathrm{\Delta }{\lambda }_{v}|+|{\lambda }_{n}||{\phi }_{n}|{w}_{n}^{\alpha }\\ =& {T}_{n,1}^{\alpha }+{T}_{n,2}^{\alpha }+{T}_{n,3}^{\alpha }.\end{array}\hfill \end{array}$

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 ${k}^{\prime }$, where $\frac{1}{k}+\frac{1}{{k}^{\prime }}=1$, we get that

by virtue of the hypotheses of the theorem and Lemma 2. Again, we get that

by hypotheses of the theorem and Lemma 2. Finally, as in ${T}_{n,1}^{\alpha }$, we have that

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take $k=1$, then we get a new result concerning the $|C,\alpha |$ summability factors of infinite series.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

The author expresses his thanks to the referees for their useful comments and suggestions.

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Correspondence to Hüseyin Bor.

### Competing interests

The author declares that he has no competing interests.

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Bor, H. Almost increasing sequences and their new applications. J Inequal Appl 2013, 207 (2013). https://doi.org/10.1186/1029-242X-2013-207

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• DOI: https://doi.org/10.1186/1029-242X-2013-207

### Keywords

• increasing sequences
• Cesàro mean
• summability factors
• Hölder inequality
• Minkowski inequality 