Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 3 (2012.), Pages 29-33
ON A NEW APPLICATION OF ALMOST INCREASING
SEQUENCES
(COMMUNICATED BY H. BOR)
W.T. SULAIMAN
Abstract. A new result concerning absolute summability of infinite series
using almost increasing sequence is presented. An application gives some generalization of Bor’s result [1].
1. Introduction
α
Let
an be an infinite series with sequence of partial sums (sn ). By uα
n , tn
we denote the nth Cesaro mean of order α > −1 of the sequences (sn ), (nan )
respectively, that is
n
1 X α−1
uα
An−v sv ,
(1.1)
=
n
Aα
n v=0
P
tα
n =
The series
P
n
1 X α−1
An−v vav .
Aα
n v=0
(1.2)
an is summable |C, α|k , k ≥ 1, if
∞
X
n=1
∞
X
k
α k
nk−1 uα
−
u
≡
n−1 |tα
n
n−1
n | < ∞.
(1.3)
n=1
For α = 1, |C, α|k summability reduces to |C, 1|k summability.
Let (pn ) be a sequence of constants such that
Pn = p0 + p1 + ... + pn → ∞ as n → ∞.
The sequence to sequence transformation
δn =
n
1 X
pn−v sv
Pn v=0
(1.4)
2000 Mathematics Subject Classification. 40D15, 40F05, 40G05.
Key words and phrases. Almost increasing sequence, Nörlund summability, absolute
summability.
c
2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted March 1, 2012. Published July 15, 2012.
29
30
W.T. SULAIMAN
defines the sequence (δn ) of the Nörlund
mean of the sequence (sn ) generated by
P
the sequence of coefficients (pn ).
an is said to be summable |N, pn |k , k ≥ 1, if
∞
X
k
nk−1 |δn − δn−1 | < ∞
(1.5)
Γ(n + α)
,
Γ(α)Γ(n + 1)
(1.6)
n=1
In the special case when
pn =
α ≥ 0,
|N, pn |k summability reduces to |C, α|k summability.
A positive sequence (bn ) is said to be almost increasing sequence if there exists
a positive increasing sequence (cn ) and two positive constants M and N such that
M cn ≤ bn ≤ N cn . Every increasing sequence is almost increasing, but the converse
n
need not to be true, see for example when bn = ne(−1) .
The following results are known:
Theorem
1.1. [3] Let p0 > 0, pn ≥ 0 and
sequence. If
P
P(pn ) be a non-increasing
an is summable |C, 1|k , then the series
an Pn (n + 1)−1 is summable |N, pn |k ,
k ≥ 1.
Theorem 1.2. [1] Let (pn ) be as in Theorem 1.1 and (Xn ) be almost increasing
sequence. If the conditions
∞
X
n ∆2 λn Xn < ∞,
(1.7)
n=1
|λn | Xn = O(1) as
n → ∞,
(1.8)
n
X
1
|tv |k = O(Xn ), as n → ∞,
(1.9)
v
v=1
P
are satisfied, then the series
an Pn λn (n + 1)−1 is summable |N, pn |k , k ≥ 1.
Lemma 1.3. [2] Under the conditions (1.7) and (1.8), we have
nXn |∆λn | = O(1), as
∞
X
n → ∞,
Xn |∆λn | < ∞.
(1.10)
(1.11)
n=1
2. Results
We state and prove the following result
Theorem 2.1. Let (pn ) be as in Theorem 1.1 and (Xn ) be almost increasing sequence. If the conditions (1.7), (1.8) and
ϕv = O(1),
v∆ϕv = O(1),
n
X
as
as
v → ∞,
v → ∞,
1
k
|t | = O(Xn ), as n → ∞,
k−1 v
vX
v
v=1
P
are satisfied, then the series
an λn ϕn is summable |C, 1|k , k ≥ 1.
(2.1)
(2.2)
(2.3)
ON A NEW APPLICATION OF ALMOST INCREASING SEQUENCES
31
Proof. Let Tn be the nth (C, 1) mean of the sequence (nan λn ϕn ). Therefore
n
1 X
vav λv ϕv .
n + 1 v=1
Tn =
Abel’s transformation gives
Tn
=
=
1
n+1
1
n+1
n−1
X
∆ (λv ϕv )
v=1
n−1
X
v
X
rar + λn ϕn
r=1
n
X
vav
v=1
(v + 1)tv ∆ϕv λv +
v=1
n−1
X
!
(v + 1)tv ϕv+1 ∆λv
v=1
= Tn1 + Tn2 + Tn3 .
!
+ tn ϕn λn
In order to complete the proof, by Minkowski’s inequality, it is sufficient to show
that
∞
X
1
k
|Tnj | < ∞,
n
n=1
j = 1, 2, 3.
Applying Hölder’s inequality, we have
m+1
X
n=2
1
k
|Tn1 |
n
=
m+1
X
n=2
=
k
n−1
1 1 X
(v + 1)tv ∆ϕv λv n n + 1 v=1
O(1)
m+1
X
1
nk+1
n=2
=
O(1)
m+1
X
=
O(1)
k
k
v |tv | |∆ϕv | |λv |
=
=
O(1)
O(1)
v=1
m
X
v=1
=
O(1)
O(1)
k
1
!k−1
1
n2
k
|tv |k
|λv |
vXvk−1
m−1
X
m−1
X
v=1
m+1
X
n=v+1
v −1 |tv | |λv |
n−1
X
v=1
v k |tv |k |∆ϕv |k |λv |k
v=1
=
k
v=1
v=1
m
X
k
n−1
1 X k
v |tv |k |∆ϕv |k |λv |k
n2 v=1
n=2
m
X
n−1
X
|∆λv |
v
m
X
X
|tr |k
|tv |k
+
O(1)
|λ
|
m
rXrk−1
vXvk−1
r=1
v=1
Xv |∆λv | + O(1)Xm |λm | = O(1).
32
m+1
X
n=2
W.T. SULAIMAN
1
|Tn2 |k
n
=
m+1
X
n=2
k
n−1
1 1 X
(v + 1)tv ϕv+1 ∆λv n n + 1 v=1
= O(1)
m+1
X
n=2
= O(1)
m
X
1
nk+1
vk
v=1
= O(1)
= O(1)
k
k
|tv | |ϕv+1 | |∆λv |
vk
Xvk−1
v=1
n−1
X
Xv |∆λv |
v=1
!k−1
m+1
k
|tv | |∆λv | X
1
k−1
k+1
n
Xv
n=v+1
m
k
X
|tv |
v |∆λv |
vXvk−1
v=1
m−1
X
v
m
k
k
X
X
|tr |
|tv | |∆λv |
+ O(1)m |∆λm |
|∆ (v |∆λv |)|
k−1
rXr
vXvk−1
r=1
v=1
m−1
X
m−1
X
v ∆2 λv Xv + O(1)
Xv |∆λv | + O(1)m |∆λm | Xm
v=1
= O(1)
n−1
X
v=1
v=1
= O(1).
m
X
1
k
|Tn3 |
n
n=1
=
m
X
1
k
|tn ϕn λn |
n
n=1
m
k
X
|tn |
= O(1)
|λn |
k−1
nXn
n=1
= O(1), as in the case of Tn1 .
3. Remarks
Remark 3.1. (a) It may be mentioned that condition (2.3)is weaker than (1.9). In
fact is (1.9) is satisfied, then
X
m
m
X
|tn |k
1
1
k
=
O
|tn | = O(Xm ),
k−1
k−1
n
nX
X
n
1
n=1
n=1
ON A NEW APPLICATION OF ALMOST INCREASING SEQUENCES
while if (2.3) is satisfied then,
m
m
X
X
1
1
k
k
|tn | =
|t | Xnk−1
k−1 n
n
nX
n
n=1
n=1
!
m−1
n
k
X X
|tv |
∆Xnk−1 +
=
k−1
vX
v
n=1
v=1
= O(1)
m−1
X
n=1
m
k
X
|tn |
nXnk−1
n=1
!
33
k−1
Xm
k−1
Xn ∆Xnk−1 + O(Xm )Xm
= O(Xm−1 )
m−1
X
n=1
k−1
k
Xn+1
− Xnk−1 + O(Xm
)
k−1
k
= O(Xm−1 ) Xm
− X1k−1 + O(Xm
)
k
= O(Xm
).
or we can deal with this case as follows
m
m
m
X
X
X
1
1
1
k
k
k
k−1
k−1
k
|t
|
X
=
O(X
)
|t | = O(Xm
).
|tn | =
n
n
m
k−1
k−1 n
n
nX
nX
n
n
n=1
n=1
n=1
Therefore (1.9) implies (2.3) but not conversely.
(b) The other advantage of condition (2.3) is that this condition leave no losing
through estimation concerning powers of |λn | . As an example through the proof of
k
k−1
Theorem 1.2, ıt has been substituted |λn | = |λn |
|λn | = O(|λn |), which implies
that |λn |k−1 has been lost.
Remark 3.2. By putting ϕn = Pn /(n + 1) in Theorem 2.1, we obtain Theorem 1.2
via Theorem 1.1, as follows:
As (pn ) is non-increasing, then Pn ≤ (n + 1)p0 which implies ϕn = O(1). Also
Pn
Pn+1
Pn
Pn
Pn+1
Pn
Pn+1
n∆ϕn = n
−
=n
−
−
=n
−n
n+1 n+2
n+1 n+2 n+2
(n + 1) (n + 2)
n+2
Pn
= O
+ O(pn+1 ) = O(1).
n+1
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
References
[1] H. Bor, On a new application of almost increasing sequences,,to be published in Mathem and
Computer Modelling, 35 (2011),230-233.
[2] S. M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J., 39 (1999),
67-73.
[3] R. S. Varma, On the absolute Nörlund summability factors, Riv. Mat. Univ. Parma (4),
3(1977)27-33.
Waad Sulaiman, Department of Computer Engineering, College of Engineering, University of Mosul, Iraq.
E-mail address: waadsulaiman@hotmail.com