# A new application of quasi power increasing sequences. I

## Abstract

In (Rocky Mt. J. Math. 38:801-807, 2008), we proved a theorem dealing with an application of quasi-σ-power increasing sequences. In the present paper, we prove that theorem under less and more weaker conditions. This theorem also includes some new and known results.

MSC:40D15, 40F05, 40G99, 46A45.

## 1 Introduction

A positive sequence $\left({b}_{n}\right)$ is said to be almost increasing if there exists a positive increasing sequence ${c}_{n}$ and two positive constants A and B such that $A{c}_{n}\le {b}_{n}\le B{c}_{n}$ (see [1]). A sequence $\left({\lambda }_{n}\right)$ is said to be of bounded variation, denoted by $\left({\lambda }_{n}\right)\in \mathcal{BV}$, if ${\sum }_{n=1}^{\mathrm{\infty }}|\mathrm{\Delta }{\lambda }_{n}|={\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_{n}-{\lambda }_{n+1}|<\mathrm{\infty }$. A positive sequence $X=\left({X}_{n}\right)$ is said to be a quasi-σ-power increasing sequence if there exists a constant $K=K\left(\sigma ,X\right)\ge 1$ such that $K{n}^{\sigma }{X}_{n}\ge {m}^{\sigma }{X}_{m}$ holds for all $n\ge m\ge 1$. It should be noted that every almost increasing sequence is a quasi-σ-power increasing sequence for any nonnegative σ, but the converse may not be true as can be seen by taking an example, say ${X}_{n}={n}^{-\sigma }$ for $\sigma >0$ (see [2]). Let $\left({\phi }_{n}\right)$ be a sequence of complex numbers and let $\sum {a}_{n}$ be a given infinite series with partial sums $\left({s}_{n}\right)$. We denote by ${z}_{n}^{\alpha }$ and ${t}_{n}^{\alpha }$ the n th Cesàro means of order α, with $\alpha >-1$, of the sequences $\left({s}_{n}\right)$ and $\left(n{a}_{n}\right)$, respectively, that is,

${z}_{n}^{\alpha }=\frac{1}{{A}_{n}^{\alpha }}\sum _{v=0}^{n}{A}_{n-v}^{\alpha -1}{s}_{v},$
(1)
${t}_{n}^{\alpha }=\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v},$
(2)

where

(3)

The series $\sum {a}_{n}$ is said to be summable $\phi -{|C,\alpha |}_{k}$, $k\ge 1$ and $\alpha >-1$, if (see [3])

$\sum _{n=1}^{\mathrm{\infty }}{|{\phi }_{n}\left({z}_{n}^{\alpha }-{z}_{n-1}^{\alpha }\right)|}^{k}=\sum _{n=1}^{\mathrm{\infty }}{n}^{-k}{|{\phi }_{n}{t}_{n}^{\alpha }|}^{k}<\mathrm{\infty }.$
(4)

In the special case, if we take ${\phi }_{n}={n}^{1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_{k}$ summability is the same as ${|C,\alpha |}_{k}$ summability (see [4]). Also, if we take ${\phi }_{n}={n}^{\delta +1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_{k}$ summability reduces to ${|C,\alpha ;\delta |}_{k}$ summability (see [5]).

## 2 Known result

In [6], we have proved the following theorem.

Theorem A Let $\left({\lambda }_{n}\right)\in \mathcal{BV}$ and let $\left({X}_{n}\right)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). Suppose also that there exist sequences $\left({\beta }_{n}\right)$ and $\left({\lambda }_{n}\right)$ such that

$|\mathrm{\Delta }{\lambda }_{n}|\le {\beta }_{n},$
(5)
(6)
$\sum _{n=1}^{\mathrm{\infty }}n|\mathrm{\Delta }{\beta }_{n}|{X}_{n}<\mathrm{\infty },$
(7)
(8)

If there exists an $ϵ>0$ such that the sequence $\left({n}^{ϵ-k}{|{\phi }_{n}|}^{k}\right)$ is non-increasing and if the sequence $\left({w}_{n}^{\alpha }\right)$ defined by (see [7])

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

satisfies the condition

(10)

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

It should be remarked that we have added the condition ‘$\left({\lambda }_{n}\right)\in \mathcal{BV}$’ in the statement of Theorem A because it is necessary.

## 3 The main result

The aim of this paper is to prove Theorem A under less and weaker conditions. Now, we will prove the following theorem.

Theorem Let $\left({X}_{n}\right)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). If there exists an $ϵ>0$ such that the sequence $\left({n}^{ϵ-k}{|{\phi }_{n}|}^{k}\right)$ is non-increasing and if the conditions from (5) to (8) are satisfied and if the condition

(11)

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

Remark It should be noted that condition (11) is the same as condition (10) when $k=1$. When $k>1$, condition (11) is weaker than condition (10), but the converse is not true. As in [8] we can show that if (10) is satisfied, then we get that

$\sum _{n=1}^{m}\frac{{\left(|{\phi }_{n}|{w}_{n}^{\alpha }\right)}^{k}}{{n}^{k}{X}_{n}^{k-1}}=O\left(\frac{1}{{X}_{1}^{k-1}}\right)\sum _{n=1}^{m}\frac{{\left(|{\phi }_{n}|{w}_{n}^{\alpha }\right)}^{k}}{{n}^{k}}=O\left({X}_{m}\right).$

If (11) is satisfied, then for $k>1$ we obtain that

$\sum _{n=1}^{m}\frac{{\left(|{\phi }_{n}|{w}_{n}^{\alpha }\right)}^{k}}{{n}^{k}}=\sum _{n=1}^{m}{X}_{n}^{k-1}\frac{{\left(|{\phi }_{n}|{w}_{n}^{\alpha }\right)}^{k}}{{n}^{k}{X}_{n}^{k-1}}=O\left({X}_{m}^{k-1}\right)\sum _{n=1}^{m}\frac{{\left(|{\phi }_{n}|{w}_{n}^{\alpha }\right)}^{k}}{{n}^{k}{X}_{n}^{k-1}}=O\left({X}_{m}^{k}\right)\ne O\left({X}_{m}\right).$

Also, it should be noted that the condition ‘$\left({\lambda }_{n}\right)\in \mathcal{BV}$’ has been removed.

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

Lemma 1 [9]

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

$|\sum _{p=0}^{v}{A}_{n-p}^{\alpha -1}{a}_{p}|\le \underset{1\le m\le v}{max}|\sum _{p=0}^{m}{A}_{m-p}^{\alpha -1}{a}_{p}|.$
(12)

Lemma 2 [2]

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

(13)
$\sum _{n=1}^{\mathrm{\infty }}{\beta }_{n}{X}_{n}<\mathrm{\infty }.$
(14)

## 4 Proof of the theorem

Let $\left({T}_{n}^{\alpha }\right)$ be the n th $\left(C,\alpha \right)$, with $0<\alpha \le 1$, mean of the sequence $\left(n{a}_{n}{\lambda }_{n}\right)$. Then, by (2), we have

${T}_{n}^{\alpha }=\frac{1}{{A}_{n}^{\alpha }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}v{a}_{v}{\lambda }_{v}.$
(15)

First, applying Abel’s transformation and then using Lemma 1, we get that

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

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. Finally, we have that

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take $ϵ=1$ and ${\phi }_{n}={n}^{1-\frac{1}{k}}$ (resp. $ϵ=1$, $\alpha =1$ and ${\phi }_{n}={n}^{1-\frac{1}{k}}$), then we get a new result dealing with ${|C,\alpha |}_{k}$ (resp. ${|C,1|}_{k}$) summability factors. Also, if we set $ϵ=1$ and ${\phi }_{n}={n}^{\delta +1-\frac{1}{k}}$, then we get another new result concerning the ${|C,\alpha ;\delta |}_{k}$ summability factors. Finally, if we take $\left({X}_{n}\right)$ as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see [10]).

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Correspondence to Hüseyin Bor.

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Bor, H. A new application of quasi power increasing sequences. I. J Inequal Appl 2013, 69 (2013). https://doi.org/10.1186/1029-242X-2013-69