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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ1_HTML.gif)
A series , with partial sums
, is said to be summable
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ2_HTML.gif)
and it is said to be summable and
if (see, [2])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_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, [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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ7_HTML.gif)
If is a quasi
-increasing sequence and condition (xi) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ10_HTML.gif)
If then there exists a natural number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ11_HTML.gif)
for all
Lemma 3.2 (see [9]).
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%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ13_HTML.gif)
where imply conditions (iv) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ16_HTML.gif)
4. Proof of Theorem 2.1
Proof.
Let be the
th term of the A transform of the partial sums of
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ17_HTML.gif)
and, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ18_HTML.gif)
We may write (noting that (vii) implies that ),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ19_HTML.gif)
To complete the proof it is sufficient, by Minkowski's inequality, to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ20_HTML.gif)
From the definition of and using (vi) and (vii) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ21_HTML.gif)
Using Hölder's inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ22_HTML.gif)
Thus, using (vii),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ23_HTML.gif)
Since is bounded by Lemma 3.3, using (v), (ix), (xi), (i), and condition (3.7) of Lemma 3.4
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ24_HTML.gif)
Using Hölder's inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ25_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%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ27_HTML.gif)
We have, using (v) and (x),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ28_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ29_HTML.gif)
Using summation by parts, (2.2), (xi), and condition (3.6) and (3.7) of Lemma 3.4
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ30_HTML.gif)
Using Hölder's inequality and (viii),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ31_HTML.gif)
Using boundedness of , (v), (x), (xi), Lemmas 3.3 and 3.4
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ32_HTML.gif)
Using summation by parts
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ33_HTML.gif)
Finally, using boundedness of , and (v) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F105136/MediaObjects/13660_2010_Article_2052_Equ37_HTML.gif)
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