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# Notes on Summability Factors of Infinite Series

*Journal of Inequalities and Applications*
**volume 2011**, Article number: 365453 (2011)

## Abstract

New result concerning summability of the infinite series is presented.

## 1. Introduction

Let be a given infinite series with sequence of partial sums . Let denote the sequence of means of . The transform of is defined by

where

Necessary and sufficient conditions for the method to be regular are

(i) for each *,*

(ii), where is a positive constant independent of *.*

The series is said to be summable , , if

where

where as .

The series is said to be summable , if

where

and it is said to be summable , , if

where is as defined by (1.1).

For , summability reduces to summability.

The series is said to be bounded or if

By , we denote the set of sequences satisfying

It is known (Das [1]) that for , (1.5) holds if and only if

For , the series is said to be -summable, , (Sulaiman [2]), if

where as .

It is quite reasonable to give the following definition.

For , the series is said to be -summable, , if

where as .

We also assume that , are positive sequences of numbers such that

A positive sequence is said to be a quasi--power increasing sequence, , if there exists a constant such that

holds for (see [3]).

Das [1], in 1966, proved the following result.

Theorem 1.1.

Let , . Then if is -summable, it is -summable.

Recently Singh and Sharma [4] proved the following theorem.

Theorem 1.2.

Let , and let be a monotonic nondecreasing sequence for . The necessary and sufficient condition that is -summable whenever

is that

## 2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,

Proof.

Since is nonincreasing, then .

Therefore

Lemma 2.2.

For ,

Proof.

Since , then is nonincreasing and hence

Lemma 2.3 (see [3]).

If is a quasi-f-increasing sequence, where , , , then under the conditions

one has

## 3. Result

Our aim is to present the following new general result.

Theorem 3.1.

Let , and let be a quasi-f-increasing sequence, where , , and (2.6), and

are all satisfied, then the series is summable , .

Proof.

We have

In order to prove the result, it is sufficient, by Minkowski's inequality, to show that

Applying HÖlder's inequality, we have

This completes the proof of the theorem.

## References

Das G:

**On some methods of summability.***The Quarterly Journal of Mathematics Oxford Series*1966,**17**(2):244–256.Sulaiman WT:

**Notes on two summability methods.***Pure and Applied Mathematika Sciences*1990,**31**(1–2):59–69.Sulaiman WT:

**Extension on absolute summability factors of infinite series.***Journal of Mathematical Analysis and Applications*2006,**322**(2):1224–1230. 10.1016/j.jmaa.2005.09.019Singh N, Sharma N:

**On****summability factors of infinite series.***Proceedings of Mathematical Sciences*2000,**110**(1):61–68. 10.1007/BF02829481

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Cite this article

Sulaiman, W.T. Notes on Summability Factors of Infinite Series.
*J Inequal Appl* **2011**, 365453 (2011). https://doi.org/10.1155/2011/365453

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DOI: https://doi.org/10.1155/2011/365453

### Keywords

- Positive Constant
- General Result
- Infinite Series
- Positive Sequence
- Nondecreasing Sequence