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A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability
Journal of Inequalities and Applications volume 2009, Article number: 675403 (2009)
Abstract
We prove two theorems on ,
, summability factors for an infinite series by using quasi-power increasing sequences. We obtain sufficient conditions for
to be summable
,
, by using quasi-f-increasing sequences.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ1_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ2_HTML.gif)
In [3], Flett considered further extension of absolute summability in which he introduced a further parameter The series
is said to be summable
,
,
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ6_HTML.gif)
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.
Let be a lower triangular matrix with nonnegative entries satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ12_HTML.gif)
and let and
be sequences such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ14_HTML.gif)
If is a quasi-
-power increasing sequence for some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ17_HTML.gif)
then the series is summable
,
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.
Let satisfy conditions (1.7)–(1.12), and let
and
be sequences satisfying conditions (1.13) and (1.14) of Theorem 1.1 and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ18_HTML.gif)
If is a quasi-
-increasing sequence and conditions (1.17) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ19_HTML.gif)
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.
Let satisfy conditions (1.7)–(1.12), 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 (1.17) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ20_HTML.gif)
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]).
Let be a sequence of real numbers and denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ21_HTML.gif)
If then there exists a natural number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ22_HTML.gif)
for all
Lemma 3.2 (see [8]).
If is a quasi-
-increasing sequence, where
then conditions (2.1) of Theorem 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ24_HTML.gif)
where imply conditions (1.15) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ25_HTML.gif)
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.
Let be a quasi-
-increasing sequence, where
If conditions (1.13), (1.14), and (2.2) are satisfied, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ27_HTML.gif)
Proof.
It is clear that if is quasi-
-increasing, then
is quasi-increasing. Since
from the fact that
is increasing and (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ28_HTML.gif)
Again using (2.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ29_HTML.gif)
4. Proof of Theorem 2.1
Let denote the
th term of the
-transform of the series
Then, by definition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ30_HTML.gif)
Then, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ31_HTML.gif)
Applying Abel's transformation, we may write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ32_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ33_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ34_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ35_HTML.gif)
to complete the proof, it is sufficient to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ36_HTML.gif)
Since is bounded by Lemma 3.3, using (1.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ37_HTML.gif)
Using properties (1.15), in view of Lemma 3.3, and (3.7), from (1.9), (1.13), and (1.17),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ38_HTML.gif)
Applying Hölder's inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ39_HTML.gif)
Using (1.9) and (1.11) and boundedness of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ40_HTML.gif)
as in the proof of
Finally, again using Hölder's inequality, from (1.9), (1.10), and (1.12),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ41_HTML.gif)
By Lemma 3.1, condition (3.3), in view of Lemma 3.3, implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ42_HTML.gif)
holds. Thus, by Lemma 3.3, (3.4) implies that is bounded. Therefore, from (1.9) and (1.13),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ43_HTML.gif)
Using Abel transformation and (1.17),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ44_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ45_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ46_HTML.gif)
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.
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-
-increasing sequence, where
and conditions (2.2) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ47_HTML.gif)
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.
Let be a positive sequence satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F675403/MediaObjects/13660_2009_Article_1988_Equ49_HTML.gif)
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
,
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Savaş, E., Şevli, H. A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability. J Inequal Appl 2009, 675403 (2009). https://doi.org/10.1155/2009/675403
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DOI: https://doi.org/10.1155/2009/675403