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A Summability Factor Theorem for Quasi-Power-Increasing Sequences
Journal of Inequalities and Applications volume 2010, Article number: 105136 (2010)
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
A series , with partial sums , is said to be summable if
and it is said to be summable and if (see, [2])
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
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:
where
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
If is a quasi -increasing sequence and condition (xi) and
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
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
If then there exists a natural number such that
for all
Lemma 3.2 (see [9]).
If is a quasi -increasing sequence, where then conditions (2.1) of Theorem 2.1,
where imply conditions (iv) and
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
4. Proof of Theorem 2.1
Proof.
Let be the th term of the A transform of the partial sums of . Then we have
and, for , we have
We may write (noting that (vii) implies that ),
To complete the proof it is sufficient, by Minkowski's inequality, to show that
From the definition of and using (vi) and (vii) it follows that
Using Hölder's inequality
Thus, using (vii),
Since is bounded by Lemma 3.3, using (v), (ix), (xi), (i), and condition (3.7) of Lemma 3.4
Using Hölder's inequality,
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 converges. Therefore, there exists a positive constant such that and from the properties of matrix , we obtain
We have, using (v) and (x),
Therefore,
Using summation by parts, (2.2), (xi), and condition (3.6) and (3.7) of Lemma 3.4
Using Hölder's inequality and (viii),
Using boundedness of , (v), (x), (xi), Lemmas 3.3 and 3.4
Using summation by parts
Finally, using boundedness of , and (v) we have
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
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
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 .
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Savaş, E. A Summability Factor Theorem for Quasi-Power-Increasing Sequences. J Inequal Appl 2010, 105136 (2010). https://doi.org/10.1155/2010/105136
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DOI: https://doi.org/10.1155/2010/105136