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Abstract and Applied Analysis
Volume 2012, Article ID 436094, 6 pages
doi:10.1155/2012/436094
Research Article
Power Increasing Sequences and Their Some
New Applications
Hüseyin Bor
P.O. Box 121, Bahçelievler, 06502 Ankara, Turkey
Correspondence should be addressed to Hüseyin Bor, hbor33@gmail.com
Received 10 August 2012; Revised 29 August 2012; Accepted 12 September 2012
Academic Editor: Toka Diagana
Copyright q 2012 Hüseyin Bor. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
In the work of Bor 2008, we have proved a result dealing with |N, pn , θn |k summability factors
by using a quasi-β-power increasing sequence. In this paper, we prove that result under less and
more weaker conditions. Some new results have also been obtained.
1. Introduction
A positive sequence bn is said to be almost increasing if there exists a positive increasing
sequence cn and two positive constants A and B such that Acn ≤ bn ≤ Bcn see 1. We write
BVO BV ∩CO , where CO {x xk ∈ Ω : limk |xk | 0}, BV{x xk ∈ Ω : k |xk −xk1 | <
∞} and Ω being the space of all real or complex-valued sequences. A positive sequence X Xn is said to be a quasi-β-power increasing sequence if there exists a constant K Kβ, X ≥
1 such that Knβ Xn ≥ mβ Xm holds for all n ≥ m ≥ 1. It should be noted that every almost
increasing sequence is a quasi-β-power increasing sequence for any nonnegative β, but the
converse is not true for β > 0. Moreover, for any positive β there exists a quasi-β-power
increasing sequence tending to infinity, but it is not almost increasing see 2. Let an be a
given infinite series with partial sums sn . Let pn be a sequence of positive numbers such
that
n
pv −→ ∞ as n −→ ∞,
P−i p−i 0, i ≥ 1 .
Pn 1.1
v0
Let θn be any sequence of positive real constants. The series
|N, pn |k , k ≥ 1, if see 3
∞ Pn k−1
|Vn − Vn−1 |k < ∞,
p
n
n1
an is said to be summable
1.2
2
Abstract and Applied Analysis
and it is said to be summable |N, pn , θn |k , k ≥ 1, if see 4
∞
θnk−1 |Vn − Vn−1 |k < ∞,
n1
where
Vn n
1
pv sv .
Pn v0
1.3
1.4
If we take θn Pn /pn , then |N, pn , θn |k summability reduces to |N, pn |k summability. Also
if we take θn n and pn 1 for all values of n, then we get |C, 1|k summability see 5.
Furthermore, if we take θn n, then |N, pn , θn |k summability reduces to |R, pn |k summability
see 6.
2. Known Result
In 7, we have proved the following theorem dealing with |N, pn , θn |k summability factors
of infinite series.
Theorem 2.1. Let λn ∈ BV O , Xn be a quasi-β-power increasing sequence for some β 0 < β < 1,
and let θn pn /Pn be a nonincreasing sequence. Suppose also there exists sequences λn and pn such that
|λm |Xm O1 as m −→ ∞,
m
nXn Δ2 λn O1,
n1
m
Pn
n1
n
2.1
OPm as m −→ ∞.
If
m
|tn |k
n1
m
n1
are satisfied, then the series
of the sequence nan .
θnk−1
n
pn
Pn
OXm as m −→ ∞,
k
|tn |k OXm as m −→ ∞,
2.2
2.3
an λn is summable |N, pn , θn |k , k ≥ 1, where tn is the nth C, 1 mean
Remark 2.2. It should be noticed that, if we take Xn as an almost increasing sequence and
θn Pn /pn , then we obtain a theorem of Mazhar see 8, in this case the condition “λn ∈
BVO ” is not needed.
3. The Main Result
The aim of this paper is to prove Theorem 2.1 under less and more weaker conditions. Now,
we prove the following theorem.
Abstract and Applied Analysis
3
Theorem 3.1. Let Xn be a quasi-β-power increasing sequence for some β 0 < β < 1, and let
θn pn /Pn be a nonincreasing sequence. Suppose also there exists sequences λn and pn such that
conditions 2.1 of Theorem 2.1 are satisfied. If
m
|tn |k
k−1
n1 nXn
OXm as m −→ ∞,
k
m
pn
|tn |k
θnk−1
OXm Pn
Xnk−1
n1
are satisfied, then the series
as m −→ ∞
3.1
3.2
an λn is summable |N, pn , θn |k , k ≥ 1.
Remark 3.2. It should be noted that conditions 3.1 and 3.2 are the same as conditions 2.2
and 2.3, respectively, when k 1. When k > 1, conditions 3.1 and 3.2 are weaker
than conditions 2.2 and 2.3, respectively. But the converses are not true. In fact, if 2.2
is satisfied, then we get that
m
|tn |k
k−1
n1 nXn
O
m
|tn |k
1
X1k−1
n1
n
OXm .
3.3
If 3.1 is satisfied, then for k > 1, we obtain that
m
|tn |k
n1
n
m
m
|tn |k
|tn |k
k−1
k
OXm .
Xnk−1
O
X
O
Xm
/
m
k−1
nXnk−1
n1
n1 nXn
3.4
The similar argument is also valid for the conditions 2.3 and 3.2. Also it should be noted
that condition “λn ∈ BV O ” has been removed.
We need following lemma for the proof of our theorem.
Lemma 3.3 see 9. Under the conditions on the sequences Xn and λn as expressed in the
statement of the theorem, one has the following:
nXn |Δλn | O1,
∞
Xn |Δλn | < ∞.
3.5
n1
4. Proof of the Theorem
Let Tn denote the N, pn mean of the series
Tn − Tn−1 an λn . Then, for n ≥ 1, we have
n
n
pn pn Pv−1 λv
vav .
Pv−1 av λv Pn Pn−1 v1
Pn Pn−1 v1 v
4.1
4
Abstract and Applied Analysis
By Abel’s transformation, we have
Tn − Tn−1 n−1
n−1
pn pn v1
v1
n1
pn tn λn −
pv tv λv
Pv Δλv tv
nPn
Pn Pn−1 v1
v
Pn Pn−1 v1
v
n−1
pn 1
Pv tv λv1 Tn,1 Tn,2 Tn,3 Tn,4 .
Pn Pn−1 v1
v
4.2
To complete the proof of the theorem, by Minkowski’s inequality, it is enough to show that
∞
θnk−1 |Tn,r |k < ∞,
for r 1, 2, 3, 4.
4.3
n1
Firstly, we have that
k
m
m
pn
θnk−1 |Tn,1 |k θnk−1 |λn |k−1 |λn |
|tn |k
P
n
n1
n1
O1
m
n1
O1
O1
|λn |θnk−1
1
Xv
k−1 pn
Pn
k
|tn |k
k
k
n
m
pv
|tv |k
|tn |k
k−1 pn
Δ|λn | θvk−1
O1|λ
θ
|
m
n
Pv
Pn
Xvk−1
Xnk−1
n1
v1
n1
m−1
4.4
m−1
|Δλn |Xn O1|λm |Xm
n1
O1
as m −→ ∞,
by virtue of the hypotheses of the theorem and lemma. Now, when k > 1 applying Hölder’s
inequality with indices k and k , where 1/k 1/k 1, as in Tn,1 , we have that
m1
n2
θnk−1 |Tn,2 |k
O1
m1
n2
O1
O1
θnk−1
pn
Pn
k
1
Pn−1
n−1
k
pv |λv | |tv |
k
×
v1
n−1
1 Pn−1 v1
pv
m1
m
θn pn k−1 pn
pv |λv |k−1 |λv ||tv |k
Pn
Pn Pn−1
v1
nv1
m θv pv k−1
v1
Pv
pv |tv |k |λv |
1
Xv
k−1 m1
pn
P
P
nv1 n n−1
k
k−1
m
1
k−1 pv
O1 θv
|λv |
|tv |k
P
X
v
v
v1
O1
k−1
k
m
pv
|tv |k
O1
|λv |θvk−1
Pv
Xvk−1
v1
as m −→ ∞.
4.5
Abstract and Applied Analysis
5
Again we have that
m1
n2
θnk−1 |Tn,3 |k
O1
m1
n2
O1
θnk−1
m
Pv
pn
Pn
1
n−1
Pv
Pn−1
v1
k
v
|tv |k vk |Δλv |k−1 |Δλv |
k k
|Δλv | v |tv |
m1
θn pn
Pn
k
k−1
v
v1
nv1
k−1
k−1
m
θv pv
1
O1
vk−1
|Δλv ||tv |k
P
vX
v
v
v1
k−1 m
θ1 p1
|tv |k
O1
v|Δλv |
P1
vXvk−1
v1
O1
O1
m−1
v
|ti |k
v1
i1
Δv|Δλv |
k−1
pn
Pn Pn−1
m
|tv |k
4.6
k−1
v1 vXv
|Δv|Δλv ||Xv O1m|Δλm |Xm
v 1Δ2 λv − |Δλv ||Xv O1m|Δλm Xm
m−1
v1
O1
×
n−1
1 Pv
Pn−1 v1 v
m−1
v1
O1
iXik−1
O1m|Δλm |
m−1
vΔ2 λv Xv O1 |Δλv |Xv
m−1
v1
v1
O1m|Δλm |Xm O1
as m −→ ∞,
by virtue of the hypotheses of the theorem and lemma. Finally, we have that
m1
n2
θnk−1 |Tn,4 |k
≤
k−1
n−1
1
Pv
1 ×
Pv |λv1 | |tv |
Pn−1 v1
v
Pn−1 v1 v
n2
k−1
m
m1
pn
1 θn pn
O1 Pv |λv1 |k−1 |λv1 ||tv |k
v nv1 Pn
Pn Pn−1
v1
k−1
k−1
m
m1
pn
1 θn pn
1
O1 Pv
|λv1 ||tv |k
Xv
v nv1 Pn
Pn Pn−1
v1
k−1
m
θv pv
|tv |k
O1 |λv1 |
Pv
vXvk−1
v1
m
θ1 p1 k−1 |tv |k
O1
|λv1 |
P1
vXvk−1
v1
m1
θnk−1
O1
O1
k
n−1
1 k
k
m−1
v
m
|tr |k
|tv |k
Δ|λv1 |
O1|λ
|
m1
k−1
k−1
v1
r1 rXr
v1 vXv
m−1
v1
O1
pn
Pn
|Δλv1 |Xv1 O1|λm1 |Xm1
as m −→ ∞,
4.7
6
Abstract and Applied Analysis
by virtue of the hypotheses of the theorem and lemma. This completes the proof of the
theorem. If we take pn 1 for all values of n and θn n, then we get a result dealing with
|C, 1|k summability factors. Also, if we take pn 1 for all values of n, then we have a new
result for |C, 1, θn |k summability. Finally, if we take θn n, then we have another new result
for |R, pn |k summability factors.
References
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