A Note on Generalized -Summability Factors for Infinite Series

Abstract

A general theorem concerning the —summability factors of infinite series has been proved.

1. Introduction

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

Mishra and Srivastava [1] obtained sufficient conditions on a sequence and a sequence for the series to be absolutely summable by the weighted mean matrix .

Recently Savaş and Rhoades [2] established the corresponding result for a nonnegative triangle, using the correct definition of absolute summability of order .

Let be an infinite lower triangular matrix. We may associate with two lower triangular matrices and , whose entries are defined by

(1.1)

respectively. The motivation for these definitions will become clear as we proceed.

Let be an infinite matrix. The series is said to be absolutely summable by , of order , written as , if

(1.2)

where is the forward difference operator and denotes the term of the matrix transform of the sequence , where .

Thus

(1.3)

since .

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

A positive sequence is said to be an almost increasing sequence if there exist an increasing sequence and positive constants and such that (see [3]). 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

(1.4)

holds for all . It should be noted that every almost increasing sequence is 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 [4]). 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 , where , , was considered instead of (see [5, 6]).

Given any sequence , the notation means and .

Quite recently, Savaş and Rhoades [2] proved the following theorem for -summability factors of infinite series.

Theorem 1.1.

Let be a triangle with nonnegative entries satisfying

(i),

(ii)  for  ,

(iii),

(iv), and

(v).

If is a positive nondecreasing sequence and the sequences and satisfy

(vi),

(vii),

(viii),

(ix), and

(x),

then the series is summable .

It should be noted that if is an almost increasing sequence then (viii) implies that the sequence is bounded. However, when is a quasi -power increasing sequence or a quasi -increasing sequence, (viii) does not imply , For example, since is a quasi -power increasing sequence for if we take , then , holds but (see [7]).

The goal of this paper is to prove a theorem by using quasi -increasing sequences. We show that the crucial condition of our proof, can be deduced from another condition of the theorem.

2. The Main Results

We have the following theorem:

Theorem 2.1.

Let be nonnegative triangular matrix satisfying conditions (i)–(v) and let and be sequences satisfying conditions (vi) and (vii) of Theorem 1.1 and

(2.1)

If is a quasi -increasing sequence and condition () and

(2.2)

are satisfied, then the series is summable , where , , , and

Theorem 2.1 includes the following theorem with the special case .

Theorem 2.2.

Let satisfying conditions (i)–(v) and let and be sequences satisfying conditions (vi), (vii), and (2.1). If is a quasi -power increasing sequence for some and conditions () and

(2.3)

are satisfied, where then the series is summable .

If we take that is an almost increasing sequence instead of a quasi -power increasing sequence then our Theorem 2.2 reduces to [8, Theorem ].

Remark 2.3.

The crucial condition, and condition (viii) 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 the Theorem 2.1, also in the statement of Theorem 2.2 with the special case conditions and (viii) hold.

3. Lemmas

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

Lemma 3.1 (see [9]).

Let be a sequence of real numbers and denote

(3.1)

If then there exists a natural number such that

(3.2)

for all

Lemma 3.2 (see [7]).

If is a quasi -increasing sequence, where , , , then conditions (2.1) of Theorem 2.1,

(3.3)

(3.4)

where imply conditions (viii) and

(3.5)

Lemma 3.3 (see [7]).

If is a quasi -increasing sequence, where , , then under conditions (vi), (vii), (2.1) and (2.2), conditions (viii) and (3.5) are satisfied.

Lemma 3.4 (see [7]).

Let be a quasi -increasing sequence, where , , If conditions (vi), (vii), and (2.2) are satisfied, then

(3.6)

(3.7)

4. Proof of Theorem 2.1

Let denote the th term of the -transform of the partial sums of the series . Then, we have

(4.1)

Thus,

(4.2)

It is easy to see that

(4.3)

Also we may write

(4.4)

Therefore, for

(4.5)

To complete the proof of the theorem, it will be sufficient to show that

(4.6)

Using Hölder's inequality and condition (iii),

(4.7)

Since is bounded by Lemma 3.3, using (ii), (iii), (vi), (x), and property (3.7) of Lemma 3.4,

(4.8)

Now

(4.9)

From [2],

(4.10)

Thus, using (iv) and (ii),

(4.11)

Hence, using Hölder's inequality, (v), (iii), and the fact that the 's are bounded,

(4.12)

as in the proof of .

It follows from (3.6) that and hence that by condition (vi).

Using (iii), Hölder's inequality, and (v),

(4.13)

Since by (x), we have

(4.14)

Using Abel's transformation, (vi), (2.2), and properties (3.7) and (3.6) of Lemma 3.4,

(4.15)

Using the boundedness of and (x),

(4.16)

as in the proof of .

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

Corollary 4.1.

Let be a positive sequence satisfying

(i) and

(ii)

and let and be sequences satisfying conditions (vi), (vii), and (2.1). If is a quasi -increasing sequence, where , and conditions (x) and (2.2) are satisfied, then the series is summable , .