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# On generalized absolute Cesàro summability factors

Journal of Inequalities and Applications20122012:253

https://doi.org/10.1186/1029-242X-2012-253

• Accepted: 17 October 2012
• Published:

## Abstract

In this paper, a known theorem dealing with $|C,\alpha ,\gamma ;\delta {|}_{k}$ summability factors has been generalized for $|C,\alpha ,\beta ,\gamma ;\delta {|}_{k}$ summability factors. Some results have also been obtained.

MSC:40D15, 40F05, 40G99.

## Keywords

• Hölder’s inequality
• quasi-monotone sequence
• summability factors

## 1 Introduction

A sequence $\left({b}_{n}\right)$ of positive numbers is said to be quasi-monotone if $n\mathrm{\Delta }{b}_{n}\ge -\rho {b}_{n}$ for some $\rho >0$ and is said to be δ-quasi-monotone, if ${b}_{n}\to 0$, ${b}_{n}>0$ ultimately and $\mathrm{\Delta }{b}_{n}\ge -{\delta }_{n}$, where $\left({\delta }_{n}\right)$ is a sequence of positive numbers (see [1]). Let $\sum {a}_{n}$ be a given infinite series with partial sums $\left({s}_{n}\right)$. We denote by ${u}_{n}^{\alpha ,\beta }$ and ${t}_{n}^{\alpha ,\beta }$ the n th Cesàro means of order $\left(\alpha ,\beta \right)$, with $\alpha +\beta >-1$, of the sequences $\left({s}_{n}\right)$ and $\left(n{a}_{n}\right)$, respectively, that is (see [2]),
(1)
(2)
where
(3)
The series $\sum {a}_{n}$ is said to be summable $|C,\alpha ,\beta {|}_{k}$, $k\ge 1$ and $\alpha +\beta >-1$, if (see [3])
$\sum _{n=1}^{\mathrm{\infty }}{n}^{k-1}|{u}_{n}^{\alpha ,\beta }-{u}_{n-1}^{\alpha ,\beta }{|}^{k}<\mathrm{\infty }.$
(4)
Since ${t}_{n}^{\alpha ,\beta }=n\left({u}_{n}^{\alpha ,\beta }-{u}_{n-1}^{\alpha ,\beta }\right)$ (see [3]), condition (4) can also be written as
$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}|{t}_{n}^{\alpha ,\beta }{|}^{k}<\mathrm{\infty }.$
(5)
The series $\sum {a}_{n}$ is said to be summable $|C,\alpha ,\beta ,\gamma ;\delta {|}_{k}$, $k\ge 1$, $\alpha +\beta >-1$, $\delta \ge 0$ and γ is a real number, if (see [4])
$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma \left(\delta k+k-1\right)}|{u}_{n}^{\alpha ,\beta }-{u}_{n-1}^{\alpha ,\beta }{|}^{k}=\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma \left(\delta k+k-1\right)-k}|{t}_{n}^{\alpha ,\beta }{|}^{k}<\mathrm{\infty }.$
(6)

If we take $\beta =0$, then $|C,\alpha ,\beta ,\gamma ;\delta {|}_{k}$ summability reduces to $|C,\alpha ,\gamma ;\delta {|}_{k}$ summability (see [5]).

## 2 Known result

In [6], we have proved the following theorem dealing with $|C,\alpha ,\gamma ;\delta {|}_{k}$ summability factors of infinite series.

Theorem A Let $k\ge 1$, $0\le \delta <\alpha \le 1$, and γ be a real number such that $-\gamma \left(\delta k+k-1\right)+\left(\alpha +1\right)k>1$. Suppose that there exists a sequence of numbers $\left({B}_{n}\right)$ such that it is δ-quasi-monotone with $|\mathrm{\Delta }{\lambda }_{n}|\le |{B}_{n}|$, ${\lambda }_{n}\to 0$ as $n\to \mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}n{\delta }_{n}logn<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}n{B}_{n}logn$ is convergent. If the sequence $\left({w}_{n}^{\alpha }\right)$ defined by (see [7])
(7)
(8)
satisfies the condition
$\sum _{n=1}^{m}{n}^{\gamma \left(\delta k+k-1\right)-k}{\left({w}_{n}^{\alpha }\right)}^{k}=O\left(logm\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{0.5em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.5em}{0ex}}m\to \mathrm{\infty },$
(9)

then the series $\sum {a}_{n}{\lambda }_{n}$ is summable $|C,\alpha ,\gamma ;\delta {|}_{k}$.

## 3 The main result

The aim of this paper is to generalize Theorem A for $|C,\alpha ,\beta ,\gamma ;\delta {|}_{k}$ summability. We shall prove the following theorem.

Theorem Let $k\ge 1$, $0\le \delta <\alpha \le 1$, and γ be a real number such that $\left(\alpha +\beta +1-\gamma \left(\delta +1\right)\right)k>1$, and let there be sequences $\left({B}_{n}\right)$ and $\left({\lambda }_{n}\right)$ such that the conditions of Theorem A are satisfied. If the sequence $\left({w}_{n}^{\alpha ,\beta }\right)$ defined by
(10)
(11)
satisfies the condition
$\sum _{n=1}^{m}{n}^{\gamma \left(\delta k+k-1\right)-k}{\left({w}_{n}^{\alpha ,\beta }\right)}^{k}=O\left(logm\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.5em}{0ex}}m\to \mathrm{\infty },$
(12)

then the series $\sum {a}_{n}{\lambda }_{n}$ is summable $|C,\alpha ,\beta ,\gamma ;\delta {|}_{k}$. It should be noted that if we take $\beta =0$, then we get Theorem A.

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

Lemma 1 ([8])

Under the conditions on $\left({B}_{n}\right)$, as taken in the statement of the theorem, we have the following:
(13)
(14)

Lemma 2 ([9])

If $0<\alpha \le 1$, $\beta >-1$, and $1\le v\le n$, then
$|\sum _{p=0}^{v}{A}_{n-p}^{\alpha -1}{A}_{p}^{\beta }{a}_{p}|\le \underset{1\le m\le v}{max}|\sum _{p=0}^{m}{A}_{m-p}^{\alpha -1}{A}_{p}^{\beta }{a}_{p}|.$
(15)

## 4 Proof of the theorem

Let $\left({T}_{n}^{\alpha ,\beta }\right)$ be the n th $\left(C,\alpha ,\beta \right)$ mean of the sequence $\left(n{a}_{n}{\lambda }_{n}\right)$. Then by (2), we have
${T}_{n}^{\alpha ,\beta }=\frac{1}{{A}_{n}^{\alpha +\beta }}\sum _{v=1}^{n}{A}_{n-v}^{\alpha -1}{A}_{v}^{\beta }v{a}_{v}{\lambda }_{v}.$
Firstly applying Abel’s transformation and then using Lemma 2, we have that
since
$|{T}_{n,1}^{\alpha ,\beta }+{T}_{n,2}^{\alpha ,\beta }{|}^{k}\le {2}^{k}\left(|{T}_{n,1}^{\alpha ,\beta }{|}^{k}+|{T}_{n,2}^{\alpha ,\beta }{|}^{k}\right).$
(16)
In order to complete the proof of the theorem, by (6), it is sufficient to show that for $r=1,2$,
$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma \left(\delta k+k-1\right)-k}|{T}_{n,r}^{\alpha ,\beta }{|}^{k}<\mathrm{\infty }.$
Whenever $k>1$, we can apply Hölder’s inequality with indices k and ${k}^{\prime }$, where $\frac{1}{k}+\frac{1}{{k}^{\prime }}=1$, we get that
in view of the hypotheses of the theorem and Lemma 1. Similarly, we have that
by virtue of the hypotheses of the theorem and Lemma 1. Therefore, by (6), we get that for $r=1,2$,
$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma \left(\delta k+k-1\right)-k}|{T}_{n,r}^{\alpha ,\beta }{|}^{k}<\mathrm{\infty }.$

This completes the proof of the theorem. If we take $\delta =0$ and $\gamma =1$, then we get a result for $|C,\alpha ,\beta {|}_{k}$ summability factors. Also, if we take $\beta =0$, $\delta =0$, and $\alpha =1$, then we get a result for $|C,1{|}_{k}$ summability.

## Authors’ Affiliations

(1)
Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey

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