Notes on Summability Factors of Infinite Series

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

(1.1)

where

(1.2)

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

(1.3)

where

(1.4)

where as .

The series is said to be summable , if

(1.5)

where

(1.6)

and it is said to be summable , , if

(1.7)

where is as defined by (1.1).

For , summability reduces to summability.

The series is said to be bounded or if

(1.8)

By , we denote the set of sequences satisfying

(1.9)

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

(1.10)

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

(1.11)

where as .

It is quite reasonable to give the following definition.

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

(1.12)

where as .

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

(1.13)

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

(1.14)

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

(1.15)

is that

(1.16)

2. Lemmas

Lemma 2.1.

Let be nonincreasing, . Then for , ,

(2.1)

Proof.

Since is nonincreasing, then .

(2.2)

Therefore

(2.3)

Lemma 2.2.

For ,

(2.4)

Proof.

Since ,  then is nonincreasing and hence

(2.5)

Lemma 2.3 (see [3]).

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

(2.6)

one has

(2.7)

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

(3.1)

are all satisfied, then the series is summable , .

Proof.

We have

(3.2)

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

(3.3)

Applying HÖlder's inequality, we have

(3.4)

This completes the proof of the theorem.