Journal of Applied Mathematics & Bioinformatics, vol.1, no.1, 2011, 143-150
ISSN: 1792-6602 (print), 1792-6939 (online)
© International Scientific Press, 2011
On Almost Increasing Sequences For Generalized
Absolute Summability
W. T. Sulaiman1
Abstract
A general result concerning absolute summability of infinite series by quasi-power
increasing sequence is proved. Our result gives correction and improvement to the
result of Savas and Sevli [2].
Mathematics Subject Classification: 40F05, 40D25.
Keywords : Absolute summability, Quasi-power increasing sequence.
1 Introduction
Let
∑a
n
be an infinite series with partial sum ( s n ) , A denote a lower triangular
matrix . The series
∑a
n
is said to be absolutely A-summable of order k ≥ 1 , if
∞
∑n
n =1
k −1
Tn − Tn −1 < ∞,
k
where
n
Tn = ∑ anv sv .
v =0
1
Department of Computer Engineering College of Engineering University of Mosul, Iraq,
email: waadsulaiman@hotmail.com
Article Info: Revised : March 9, 2011. Published online : May 31, 2011.
144
The series
On Almost Increasing Sequences For Generalized
∑a
n
is summable A, δ k , k ≥ 1, δ ≥ 0, if
∞
∑ nδ
k + k −1
n =1
Tn − Tn −1 < ∞.
k
(1)
A positive sequence γ = (γ n ) is said to be a quasi- β -power increasing sequence if
there exists a constant K = K ( β , γ ) ≥ 1 such that
K nβ γ n ≥ mβ γ m
(2)
Holds for all n ≥ m ≥ 1. It may be mentioned that every almost increasing
sequence is a quasi - β -power increasing sequence for any nonnegative β , but the
converse need not be true.
Two lower triangular matrices A and  are associated with A as follows
n
a nv = ∑ a nr , n, v = 0,1,..... ,
(3)
r =v
aˆnv = anv − an −1,v , n = 1, 2,....,
aˆ00 = a00 = a00 .
Savas and Sevli [2] proved the following result.
Theorem1.1. Let A be a lower triangular matrix with nonnegative entries
satisfying
a n −1,v ≥ a n ,v for n ≥ v + 1,
a n 0 = 1,
n = 0,1,.... ,
na nn = Ο (1), 1 / na nn = Ο(1) , as n → ∞,
n −1
∑a
v =1
aˆ v ,n +1 = Ο (a nn ),
vv
m +1
∑ nδ
k
Δ v aˆ nv = Ο (v δ k a vv ) ,
(4)
(5)
n = v +1
m +1
∑ n δ aˆ
k
n = v +1
n ,v +1
= Ο(v δk ) ,
and let ( β n ) and (λn ) be sequences such that
(6)
W. T. Sulaiman
145
Δλ n ≤ β n ,
(7)
β n → 0 as n → ∞ ,
(8)
If ( X n ) is a quasi - β -increasing sequence satisfying
m
∑ nδ
k −1
n =1
∞
∑ nX
n =1
n
sn = Ο ( X m ), m → ∞,
k
Δβ n < ∞,
(10)
λn X n = Ο (1) ,
then the series
∑a λ
n
n
(9)
(11)
is summable A, δ k , k ≥ 1, 0 < δ ≤ 1 / k .
We name the following condition
nδ k −1 k
s = Ο ( X m ), m → ∞ .
∑
k −1 n
n =1 X n
m
(12)
Remark 1. It may be mentioned that in the proof of theorem 1.1, an incorrect
step through the estimation of I 2 . The author consider (vβ v ) is bounded regarding
this follows from the fact vβ v X v = Ο (1). This not true, as for X n is β − quasi ,
we may take
X v = v − β , which implies via vβ v X v = Ο (1) that (vβ v ) is not
bounded.
Therefore the proof of theorem 1.1 is not valid.
2 Lemmas
Lemma 2.1. Condition (12) is weaker than (9) when X n is non-decreasing.
Proof. If (9) holds, then we have
m
sn
∑ nX
n =1
k
k −1
n
⎛ 1 ⎞ m 1
= Ο ⎜⎜ k −1 ⎟⎟ ∑ s n
⎝ X 1 ⎠ n =1 n
while if (12) is satisfied then,
k
= Ο (X m ) ,
146
On Almost Increasing Sequences For Generalized
m
1
sn
∑
n =1 n
k
=
m
1
∑ nX
n =1
s n X nk −1
k
k −1
n
⎛ m sn k ⎞
⎛ n sv k ⎞
k −1
⎟ X k −1
⎜
⎟
ΔX n + ⎜ ∑
=∑ ∑
m
k −1 ⎟
k −1 ⎟
⎜
⎜
nX n
n =1 v =1 vX v
⎝ n =1
⎠
⎝
⎠
m −1
m −1
= Ο (1) ∑ X n ΔX nk −1 + Ο ( X m ) X mk −1
n =1
m −1
(
)
( )
= Ο ( X m −1 ) ∑ X nk+−11 − X nk −1 + Ο X mk
n =1
(
)
( )
= Ο ( X m −1 ) X mk −1 − X 1k −1 + Ο X mk
( )
= Ο X mk .
Therefore (9) implies (12) but not conversely .
Remark 2.
1. Condition (9) has been replaced by (12) which is better in the following sense
(a). If X n is non-decreasing, (12) is weaker than (9) (see lemma 2.1)
(b) The more advantage of our conditions is to obtain the desired result without
any loss of powers through estimations. As an example the proof via condition
(9) impose to deal with λn
k
as λn = λn
k
k −1
λn = Ο ( λn ) , loosing λ n
k −1
as
considered to be Ο(1) . We have no such case via condition (12).
2. Condition (4) is eliminated.
Lemma 2.2. Conditions (8) and (10) imply
mX m β m = Ο (1) , m → ∞,
∞
∑β
n =1
n
X n = Ο (1) .
Proof. As β n → 0, and n β X n is non-decreasing, we have
(13)
(14)
W. T. Sulaiman
147
∞
nX n βν = n1− β n β X n ∑ Δβν
v=n
∞
= Ο (1) n1− β ∑ v β X v Δβν
v=n
∞
= Ο (1) ∑ v1− β v β X v Δβν
v=n
∞
= Ο (1) ∑ vX v Δβν = Ο (1).
v=n
This proves (13). To prove (14), we observe that
m
∑ X v βv =
v =1
⎛ m
⎞
⎛ v
⎞
Δ
+
X
β
⎜∑ X v ⎟ βm
⎜
⎟
∑
∑
r
v
v =1 ⎝ r =1
⎠
⎝ v =1
⎠
m −1
m −1
⎛ v
⎞
⎛ m
⎞
= Ο (1) ∑ ⎜ ∑ r − β r β X r ⎟ Δβ v + Ο (1) ⎜ ∑ v − β v β X v ⎟ β m
v =1 ⎝ r =1
⎠
⎝ v =1
⎠
m −1
= Ο (1) ∑ v β X v Δβ v
v =1
v
∑r
− β −∈
r∈
r =1
m
+ Ο (1) m β X m β m ∑ v − β −∈ v ∈ , ∈< 1 − β
v =1
m −1
v
v =1
r =1
= Ο (1) ∑ v β X v Δβ v v ∈ ∑ r − β −∈
m
+ Ο (1) m β X m β m m∈ ∑ v − β −∈
v =1
m
= Ο (1) ∑ v
v =1
β +∈
⎛ v − β −∈ ⎞
⎛ m − β −∈ ⎞
β +∈
X v Δβ v ⎜ ∫ u
du ⎟ + Ο (1) m X m β m ⎜ ∫ u
du ⎟
⎝1
⎠
⎝1
⎠
m
= Ο (1) ∑ v X v Δβ v + Ο (1) m X m β m
v =1
= Ο (1).
Lemma 2.3 [1]. Let A be as defined in theorem 1.1, then
aˆn ,v +1 ≤ ann for n ≥ v + 1.
148
On Almost Increasing Sequences For Generalized
3 Main Result
Suppose all conditions of theorem 1.1 are satisfied except
Theorem 3.1.
condition (9) is replaced by condition (12), and condition (4) is removed, then the
series
∑a λ
n
n
is summable A, δ k , k ≥ 1, 0 < δ ≤ 1 / k .
Proof. Let x n be the nth term of the A-transform of the series
∑a λ
n
n
definition, we have
n
n
v =0
v =0
x n = ∑ a nv s v = ∑ a nv λv a v ,
and hence
n
Tn := x n − x n −1 = ∑ aˆ nv λv a v .
v =0
Applying Abel's transformation,
n −1
n −1
v =1
v =1
Tn = ann λn sn + ∑ Δ v aˆnv λv sv + ∑ aˆn ,v +1Δλv sv = Tn1 + Tn 2 + Tn 3 .
To complete the proof, by Minkowski's inequality, it is sufficient to show that
∞
∑ nδ
k + k −1
Tnj
k
< ∞, j = 1,2,3.
n =1
Applying Holder's inequality, we have
m
∑n
n =1
δ k + k −1
m
Tn1 = ∑ n δ k + k −1 a nn λn s n
k
k
n =1
m
≤ ∑ (na nn )
n =1
k
n δ k −1
k
k −1
s n λn ( λn X n )
k −1
Xn
n δ k −1
k
= Ο (1) ∑ k −1 s n λn
n =1 X n
m
m −1
= Ο (1) ∑ Δλn
n =1
m −1
vδ k −1 k
s + Ο (1) λm
∑
k −1 v
v =1 X v
n
= Ο (1) ∑ β n X n + Ο (1) λm X m = Ο (1) .
n =1
n δ k −1
sn
∑
k −1
n =1 X n
m
k
. By
W. T. Sulaiman
m +1
∑n
δ k + k −1
n=2
149
m +1
Tn 2 = ∑ n
k
δ k + k −1
n=2
k
n −1
∑ Δ aˆ
v =1
v
λv s v
nv
n −1
n −1
m +1
⎞
k
k⎛
≤ ∑ n δ k + k −1 ∑ Δ v aˆ nv λv s v ⎜ ∑ Δ v aˆ nv ⎟
v =1
n=2
⎝ v =1
⎠
m +1
n −1
n=2
v =1
= Ο (1) ∑ n δ k (na nn ) k −1 ∑ Δ v aˆ nv λv s v
m
= Ο (1) ∑ λv sv
k
k
v =1
m
∑ nδ
n =v +1
m
k
k
Δ v aˆnv
k
= Ο (1) ∑ v δ k a vv λv s v
k
k −1
k
v =1
v δ k −1
k
k −1
s v λv ( λv X v )
k −1
v =1 X v
m
= Ο (1) ∑
v δ k −1
k
= Ο (1) ∑ k −1 s v λv
v =1 X v
m
= Ο (1), as in the case of Tn1 .
m +1
∑n
n=2
δ k + k −1
m +1
Tn 3 = ∑ n
k
δ k + k −1
n=2
m +1
≤∑n
∑ aˆ
v =1
δ k + k −1
n , v +1
Δλv s v
n −1
∑ (aˆ
v =1
n=2
k
n −1
)
k
n , v +1
m +1
n −1
n=2
v =1
Δλv s v X
k
1− k
v
⎛ n −1
⎞
⎜ ∑ Δλv X v ⎟
⎝ v =1
⎠
k −1
= Ο (1) ∑ n δ k + k −1 ∑ (aˆ n ,v +1 ) Δλv s v X v1− k
m
= Ο (1) ∑ Δλv sv X v1− k
k
v =1
m
= Ο (1) ∑ Δλv sv X v1− k
k
v =1
m
= Ο (1) ∑ Δλv sv X v1− k
v =1
k
k
k
m +1
∑ν nδ
k + k −1
aˆn ,v +1 ( aˆn ,v +1 )
k + k −1
aˆn ,v +1 ( ann )
n = +1
m +1
∑ν nδ
n = +1
m +1
∑ν nδ
n = +1
k
aˆn ,v +1 ( nann )
k −1
k −1
k −1
(by lemma 2.3)
150
On Almost Increasing Sequences For Generalized
m
= Ο (1) ∑ Δλv sv X v1− k
k
v =1
m
m +1
∑ν nδ
n = +1
k
aˆn ,v +1
= Ο (1) ∑ v δ k Δλv s v X v1− k
k
v =1
= Ο (1)
m
∑ vδ
k
v =1
m
= Ο (1) ∑ vβ v
v =1
m −1
β v s v X v1− k
k
vδ k −1
sv
X vk −1
k
m
r δ k −1
v δ k −1
k
s
+
Ο
(
1
)
m
β
sv
r
m ∑
k −1
k −1
r =1 X r
v =1 X v
v
= Ο (1) ∑ Δ(vβ v ) ∑
v =1
m
m
v =1
v =1
k
= Ο (1) ∑ β v X v + Ο (1)∑ v Δβ v X v + Ο (1) mβ m X m
= Ο (1)
ACKNOWLEDGEMENT. The author is so grateful to the referee who guided
me to improve writing of the paper.
References
[1] B. E. Rhoades, Inclusion theorems for absolute matrix summability methods,
J. Mat. Anal. Appl., 238 (1999), 87-90.
[2] E. Savas and H. Sevli, A recent note on quasi –power increasing sequence for
generalized absolute summability, Journal of Inequalities and Applications,
2009, Article ID675403, 10 pages.