A Summability Factor Theorem for Quasi-Power-Increasing Sequences

Abstract

We establish a summability factor theorem for summability , where is lower triangular matrix with nonnegative entries satisfying certain conditions. This paper is an extension of the main result of the work by Rhoades and Savaş (2006) by using quasi -increasing sequences.

1. Introduction

Recently, Rhoades and Savaş [1] obtained sufficient conditions for to be summable , by using almost increasing sequence. The purpose of this paper is to obtain the corresponding result for quasi -increasing sequence.

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

Let be a lower triangular matrix, a sequence. Then

(1.1)

A series , with partial sums , is said to be summable if

(1.2)

and it is said to be summable and if (see, [2])

(1.3)

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]). A sequence satisfying (1.4) for is 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 , (see, [5]).

We may associate with two lower triangular matrices and as follows:

(1.5)

where

(1.6)

Given any sequence , the notation means that and For any matrix entry

Rhoades and Savaş [1] proved the following theorem for summability factors of infinite series.

Theorem 1.1.

Let be an almost increasing sequence and let and be sequences such that

(i),

(ii),

(iii),

(iv)

Let be a lower triangular matrix with nonnegative entries satisfying

(v)

(vi)  for  ,

(vii)  for all  ,

(viii)

(ix) and

(x).

If

(xi), where

then the series is summable .

It should be noted that, if is an almost increasing sequence, then condition (iv) implies that the sequence is bounded. However, if is a quasi -power increasing sequence or a quasi -increasing sequence, (iv) does not imply that is bounded. For example, the sequence defined by is trivially a quasi -power increasing sequence for each If for any then but is not bounded, (see, [6, 7]).

The purpose 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 now will prove the following theorems.

Theorem 2.1.

Let satisfy conditions (v)–(x) and let and be sequences satisfying conditions (i) and (ii) of Theorem 1.1 and

(2.1)

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

(2.2)

are satisfied then the series is summable , where and

The following theorem is the special case of Theorem 2.1 for .

Theorem 2.2.

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

(2.3)

are satisfied, where then the series is summable , .

Remark 2.3.

The conditions and (iv) 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 (iv) hold.

3. Lemmas

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

Lemma 3.1 (see [8]).

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 [9]).

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

(3.3)

(3.4)

where imply conditions (iv) and

(3.5)

Lemma 3.3 (see [7]).

If is a quasi -increasing sequence, where then under conditions (i), (ii), (2.1), and (2.2), conditions (iv) and (3.5) are satisfied.

Lemma 3.4 (see [7]).

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

(3.6)

(3.7)

4. Proof of Theorem 2.1

Proof.

Let be the th term of the A transform of the partial sums of . Then we have

(4.1)

and, for , we have

(4.2)

We may write (noting that (vii) implies that ),

(4.3)

To complete the proof it is sufficient, by Minkowski's inequality, to show that

(4.4)

From the definition of and using (vi) and (vii) it follows that

(4.5)

Using Hölder's inequality

(4.6)

Thus, using (vii),

(4.7)

Since is bounded by Lemma 3.3, using (v), (ix), (xi), (i), and condition (3.7) of Lemma 3.4

(4.8)

Using Hölder's inequality,

(4.9)

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

(4.10)

holds. Thus by Lemma 3.3, (3.4) implies that converges. Therefore, there exists a positive constant such that and from the properties of matrix , we obtain

(4.11)

We have, using (v) and (x),

(4.12)

Therefore,

(4.13)

Using summation by parts, (2.2), (xi), and condition (3.6) and (3.7) of Lemma 3.4

(4.14)

Using Hölder's inequality and (viii),

(4.15)

Using boundedness of , (v), (x), (xi), Lemmas 3.3 and 3.4

(4.16)

Using summation by parts

(4.17)

Finally, using boundedness of , and (v) we have

(4.18)

as in the proof of .

5. Corollaries and Applications to Weighted Means

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

Corollary 5.1.

Let satisfy conditions (v)–(viii) and let and be sequences satisfying conditions (i), (ii), and (2.1). If is a quasi -increasing sequence, where and conditions (2.2) and

(5.1)

are satisfied then the series is summable

Proof.

If we take in Theorem 2.1 then condition (xi) reduces condition (5.1).

Corollary 5.2.

Let satisfy conditions (v)–(viii) and let and be sequences satisfying conditions (i), (ii), 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 ,

Corollary 5.3.

Let be a positive sequence such that as satisfies

(5.2)

(5.3)

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

Proof.

In Theorem 2.1, set . Conditions (i) and (ii) of Corollary 5.3 are, respectively, conditions (i) and (ii) of Theorem 2.1. Condition (v) becomes condition (5.2) and conditions (ix) and (x) become condition (5.3) for weighted mean method. Conditions (vi), (vii), and (viii) of Theorem 2.1 are automatically satisfied for any weighted mean method.

The following Corollary is the special case of Corollary 5.3 for .

Corollary 5.4.

Let be a positive sequence satisfying (5.2), (5.3) and let be a quasi -power increasing sequence for some Then under conditions (i), (ii), (xi), (2.1), and (2.3), is summable .