# A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability

- E. Savaş
^{1}Email author and - H. Şevli
^{2}

**2009**:675403

https://doi.org/10.1155/2009/675403

© E. Savaş and H. Şevli. 2009

**Received: **15 May 2009

**Accepted: **30 July 2009

**Published: **19 August 2009

## Abstract

## Keywords

## 1. Introduction

Quite recently, Savaş [1] obtained sufficient conditions for to be summable , The purpose of this paper is to obtain the corresponding result for quasi- -increasing sequence. Our result includes and moderates the conditions of his theorem with the special case .

A sequence is said to be of bounded variation if Let where denotes the set of all null sequences.

The concept of absolute summability of order was defined by Flett [2] as follows. Let denote a series with partial sums and a lower triangular matrix. Then is said to be absolutely -summable of order written that is summable if

where

In [3], Flett considered further extension of absolute summability in which he introduced a further parameter The series is said to be summable , , if

A positive sequence is said to be an almost increasing sequence if there exist an increasing sequence and positive constants and such that (see [4]). Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking the example, say .

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

holds for all . It should be noted that every almost increasing sequence is a quasi- -power increasing sequence for any nonnegative , but the converse need not be true as can be seen by taking an example, say for (see [5]). If (1.4) stays with then is simply called a quasi-increasing sequence. It is clear that if is quasi- -power increasing, then is quasi-increasing.

A positive sequence is said to be a quasi- -power increasing sequence, if there exists a constant such that holds for all , [6].

We may associate two lower triangular matrices and as follows:

where

Given any sequence the notation means and For any matrix entry

Quite recently, Savaş [1] obtained sufficient conditions for to be summable , as follows.

Theorem 1.1.

Theorem 1.1 enhanced a theorem of Savas [7] by replacing an almost increasing sequence with a quasi- -power increasing sequence for some . It should be noted that if is an almost increasing sequence, then (1.15) implies that the sequence is bounded. However, when is a quasi- -power increasing sequence or a quasi- -increasing sequence, (1.15) does not imply For example, since is a quasi- -power increasing sequence for and if we take then holds but (see [8]). Therefore, we remark that condition should be added to the statement of Theorem 1.1.

The goal of this paper is to prove the following theorem by using quasi- -increasing sequences. Our main result includes the moderated version of Theorem 1.1. We will show that the crucial condition of our proof, can be deduced from another condition of the theorem. Also, we shall eliminate condition (1.15) in our theorem; however we shall deduce this condition from the conditions of our theorem.

## 2. The Main Results

We now shall prove the following theorems.

Theorem 2.1.

are satisfied, then the series is summable where and

Theorem 2.1 includes the following theorem with the special case . Theorem 2.2 moderates the hypotheses of Theorem 1.1.

Theorem 2.2.

are satisfied, where then the series is summable ,

Remark 2.3.

The crucial condition, and condition (1.15) do not appear among the conditions of Theorems 2.1 and 2.2. By Lemma 3.3, under the conditions on and as taken in the statement of Theorem 2.1, also in the statement of Theorem 2.2 with the special case conditions and (1.15) hold.

## 3. Lemmas

We shall need the following lemmas for the proof of our main Theorem 2.1.

Lemma 3.1 (see [9]).

Lemma 3.2 (see [8]).

Lemma 3.3.

If is a quasi- -increasing sequence, where then, under conditions (1.13), (1.14), (2.1), and (2.2), conditions (1.15) and (3.5) are satisfied.

Proof.

It is clear that (1.13) and (1.14) (3.3). Also, (1.13) and (2.2) (3.4). By Lemma 3.2, under conditions (1.13)-(1.14) and (2.1)–(2.2), we have (1.15) and (3.5).

Lemma 3.4.

Proof.

## 4. Proof of Theorem 2.1

Let denote the th term of the -transform of the series Then, by definition, we have

Applying Abel's transformation, we may write

Since

we have

Since

to complete the proof, it is sufficient to show that

Since is bounded by Lemma 3.3, using (1.9), we have

Using properties (1.15), in view of Lemma 3.3, and (3.7), from (1.9), (1.13), and (1.17),

Applying Hölder's inequality,

Using (1.9) and (1.11) and boundedness of

Finally, again using Hölder's inequality, from (1.9), (1.10), and (1.12),

By Lemma 3.1, condition (3.3), in view of Lemma 3.3, implies that

holds. Thus, by Lemma 3.3, (3.4) implies that is bounded. Therefore, from (1.9) and (1.13),

Using Abel transformation and (1.17),

Since

we have

by virtue of (2.2) and properties (3.6) and (3.7) of Lemma 3.4.

So we obtain (4.7). This completes the proof.

## 5. Corollaries and Applications to Weighted Means

Setting in Theorems 2.1 and 2.2 yields the following two corollaries, respectively.

Corollary 5.1.

are satisfied, then the series is summable

Proof.

If we take in Theorem 2.1, then condition (1.17) reduces condition (5.1). In this case conditions (1.11) and (1.12) are obtained by conditions (1.7)–(1.10).

Corollary 5.2.

Let satisfy conditions (1.7)–(1.10), and let and be sequences satisfying conditions (1.13), (1.14), and (2.1). If is a quasi- -power increasing sequence for some and conditions (2.3) and (5.1) are satisfied, then the series is summable

A weighted mean matrix, denoted by is a lower triangular matrix with entries where is nonnegative sequence with and as

Corollary 5.3.

and let and be sequences satisfying conditions (1.13), (1.14), and (2.1). If is a quasi- -increasing sequence, where and conditions (1.17) and (2.2) are satisfied, then the series, is summable for and

Proof.

In Theorem 2.1 set . It is clear that conditions (1.7), (1.8), and (1.10) are automatically satisfied. Condition (1.9) becomes condition (5.2), and conditions (1.11) and (1.12) become condition (5.3) for weighted mean method.

Corollary 5.3 includes the following result with the special case

Corollary 5.4.

Let be a positive sequence satisfying (5.2) and (5.3), and let be a quasi- -power increasing sequence for some Then under conditions (1.13), (1.14), (1.17), (2.1), and (2.3), is summable ,

## Authors’ Affiliations

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