Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 365453, 8 pages
doi:10.1155/2011/365453
Research Article
Notes on |N, p, q|k Summability Factors of
Infinite Series
W. T. Sulaiman
Department of Computer Engineering, College of Engineering, University of Mosul, Mosul, Iraq
Correspondence should be addressed to W. T. Sulaiman, waadsulaiman@hotmail.com
Received 5 November 2010; Accepted 19 January 2011
Academic Editor: Paolo E. Ricci
Copyright q 2011 W. T. Sulaiman. 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.
New result concerning |N, p, q|k summability of the infinite series
an λn is presented.
1. Introduction
Let an be a given infinite series with sequence of partial sums sn . Let Tn denote the
sequence of N, p, q means of sn . The N, p, q transform of sn is defined by
Tn n
1 pn−v qv sv ,
Rn v0
1.1
where
Rn n
pn−v qv /
0,
for any n p−1 q−1 R−1 0 .
v0
Necessary and sufficient conditions for the N, p, q method to be regular are
i limn → ∞ pn−v qn /Rn 0 for each v,
ii
n
v0
|pn−v qv | < K|Rn |, where K is a positive constant independent of n.
1.2
2
The series
Journal of Inequalities and Applications
an is said to be summable |R, pn |k , k ≥ 1, if
∞
k
nk−1 ϕn − ϕn−1 < ∞,
1.3
n1
where
ϕn n
1 pv s v ,
Pn v0
1.4
where Pn p1 p2 · · · pn → ∞ as n → ∞.
The series an is said to be summable |N, pn |, if
∞
|σn − σn−1 | < ∞,
1.5
n
1 pn−v sv ,
Pn v0
1.6
n1
where
σn and it is said to be summable |N, p, q|k , k ≥ 1, if
∞
nk−1 |Tn − Tn−1 |k < ∞,
1.7
n1
where Tn is as defined by 1.1.
For k 1, |N, p, q|k summability reduces to |N, p, q| summability.
The series an is said to be N, p, q bounded or an O1N, p, q if
tn n
pn−v qv sv ORn as n −→ ∞.
1.8
v1
By M, we denote the set of sequences p pn satisfying
pn1 pn2
≤
≤ 1,
pn
pn1
pn > 0, n 0, 1, . . . .
1.9
It is known Das 1 that for p ∈ M, 1.5 holds if and only if
∞
n
1 pn−v vav < ∞.
nP
n
n1
v1
1.10
Journal of Inequalities and Applications
For p ∈ M, the series
3
an is said to be |N, pn |k -summable, k ≥ 1, Sulaiman 2, if
k
∞
n
1 p
va
< ∞,
n−v
v
k
nP
n v1
n1
1.11
where Pn p1 p2 · · · pn → ∞ as n → ∞.
It is quite reasonable to give the following definition.
For p ∈ M, the series an is said to be |N, p, q|k -summable, k ≥ 1, if
k
∞
n
1 vp
q
a
< ∞,
n−v
v
v
k
n1 nPn v1
1.12
where Pn p1 p2 · · · pn → ∞ as n → ∞.
We also assume that pn , qn are positive sequences of numbers such that
Pn p0 p1 · · · pn −→ ∞,
as n −→ ∞,
Qn q0 q1 · · · qn −→ ∞,
as n −→ ∞.
1.13
A positive sequence α αn is said to be a quasi-f-power increasing sequence, f fn , if
there exists a constant K Kα, f such that
Kfn αn ≥ fm αm ,
1.14
holds for n ≥ m ≥ 1 see 3.
Das 1, in 1966, proved the following result.
Theorem 1.1. Let pn ∈ M, qn ≥ 0. Then if
an is |N, p, q|-summable, it is |N, qn |-summable.
Recently Singh and Sharma 4 proved the following theorem.
Theorem 1.2. Let pn ∈ M, qn > 0 and let qn be a monotonic nondecreasing sequence for n ≥ 0.
The necessary and sufficient condition that an λn is |N, qn |-summable whenever
an O1 N, p, q ,
∞
qn
|λn | < ∞,
Qn
n0
∞
|Δλn | < ∞,
n0
∞
Qn1 2 Δ λn < ∞,
q
n0 n1
1.15
4
Journal of Inequalities and Applications
is that
∞
qn
|sn ||λn | < ∞.
Qn
n1
1.16
2. Lemmas
Lemma 2.1. Let pn be nonincreasing, n OPn . Then for r > 0, k ≥ 1,
∞
k
pn−v
nv1
O
nr Pnk
1
2.1
.
vrk−1
Proof. Since pn is nonincreasing, then npn OPn .
∞
k
pn−v
nv1
nr Pnk
2v
k
pn−v
nv1
nr Pnk
2v
k
pn−v
nv1
O1
O1
∞
k
pn−v
n2v1
nr Pnk
O1
nr Pnk
∞
k
pn−v
nr Pnk
n2v1
1
2v
vr Pvk
nv1
1
v
vr Pvk
m1
,
k
pn−v
O1
pm O1
∞
k
pm
r
1
v
vr Pvk
m1
1
vr Pvk−1
O1
k
pm
O
∞
,
rk−1
1
v
2.2
k
pm
k
r k
m v Pmv
mv1 m Pm
∞
∞
1
1
−r−k
.
O1
O1
x
dx
O
rk
vrk−1
v
mv1 m
mv1
Therefore
∞
k
pn−v
nv1
nr Pnk
O
1
vrk−1
.
2.3
Lemma 2.2. For p ∈ M,
∞ Δv pn−v < ∞.
2.4
v0
Proof. Since p ∈ M, then pn is nonincreasing and hence
m ∞ Δv pn−v pn−v−1 − pn−v pn − pm−v−1 O1.
v0
v0
2.5
Journal of Inequalities and Applications
5
Lemma 2.3 see 3. If Xn is a quasi-f-increasing sequence, where f fn nβ log nγ , γ > 0,
0 < β < 1, then under the conditions
Xm |λm | O1,
m
m −→ ∞,
nXn Δ2 λn O1,
m −→ ∞,
2.6
n1
one has
nXn |Δλn | O1,
∞
2.7
Xn |Δλn | < ∞.
n1
3. Result
Our aim is to present the following new general result.
Theorem 3.1. Let p ∈ M, and let Xn be a quasi-f-increasing sequence, where f fn nlogγ n,
γ > 0, 0 < β < 1 and 2.6, and
n
qv |sv |k
v1
vXvk−1
OXn ,
Δqv O v−1 qv ,
3.1
qv1 O qv ,
v OPv ,
are all satisfied, then the series
an λn is summable |N, p, q|k , k ≥ 1.
Proof. We have
Tn n
vpn−v qv av λv
v0
n−1 v
v0
n−1
v0
ar
Δv v pn−v qv λv r0
n
av np0 qn λn
v0
sv −pn−v qv λv v 1Δqv pn−v λv v 1qv1 Δv pn−v λv
v 1qv1 pn−v−1 Δλv np0 qn sn λn
Tn1 Tn2 Tn3 Tn4 Tn5 .
3.2
6
Journal of Inequalities and Applications
In order to prove the result, it is sufficient, by Minkowski’s inequality, to show that
∞
1 k
Tnj < ∞,
k
n1 nPn
j 1, 2, 3, 4, 5.
Applying HÖlder’s inequality, we have
m
1
n1
nPnk
|Tn1 |
k
k
m
n−1
1 pn−v qv sv λv k
nP
n v0
n1
≤
n−1
m
1 n1
nPnk
O1
n−1
m
Pnk−1 nPnk
m
pn−v
pn−v qvk |sv |k |λv |k
v0
qvk |sv |k |λv |k
m
k−1
v0
v0
O1
n−1
v0
n1
O1
pn−v qvk |sv |k |λv |k
∞
pn−v
nPn
nv1
v−1 qvk |sv |k |λv |k
v0
O1
m
qvk |sv |k
vXvk−1
v0
O1
m
qvk |sv |k
vXvk−1
v0
O1
m−1
Δ|λv |
v0
O1
|λv ||λv |k−1 Xvk−1
|λv |
v
qrk |sr |k
r0
rXrk−1
|λn |
m
qvk |sv |k
v0
vXvk−1
m−1
|Δλv | Xv |λm | Xm O1,
v0
k
m
n−1
1 Δq
s
λ
1p
|T
|
v
n2
n−v
v
v
v
k
k
nPn
n1 nPn v0
m
1
n1
k
≤
n−1
m
1 n1
nPnk
O1
O1
k−1
n−1
k
k
k
v pn−v Δqv |sv | |λv |
pn−v
k
v0
n−1
m
Pnk−1 k
n1 nPn v0
m
v0
k
vk pn−v Δqv |sv |k |λv |k
∞
k
pn−v
vk Δqv |sv |k |λv |k
nPn
v0
nv1
3.3
Journal of Inequalities and Applications
O1
m
7
k
vk−1 Δqv |sv |k |λv |k
v0
O1
m
qvk |sv |k
vXvk−1
v0
O1,
as in the case of Tn1 ,
k
m
n−1
1 p
q
s
λ
1Δ
|T
|
v
n3
v
n−v
v1
v
v
k
nPnk
nP
n v0
n1
m
1
n1
|λv |
k
≤
n−1
m
1 n1
nPnk
O1
O1
v0
k
n1 nPn v0
m
v0
∞ Δ p
v n−v
nPnk
nv1
m
k−1
n−1 Δv pn−v k
vk Δv pn−v qv1
|sv |k |λv |k
k
vk qv1
|sv |k |λv |k
k
vk−1 Pv−k qv1
|sv |k |λv |k
v0
O1
k
Δv pn−v qv1
|sv |k |λv |k
n−1
m
1 v0
O1
v
k
m
v−1 qvk |sv |k |λv |k
v0
O1,
k
m
n−1
1 q
s
Δλ
1p
|T
|
v
n4
n−v−1
v1
v
v
k
nPnk
nP
n v0
n1
m
1
n1
as in the case of Tn1 ,
k
≤
n−1
m
1 n1
nPnk
O1
O1
v0
n−1
m
1 k
n1 nPn v0
Xvk−1
m qk |s |k
v1 v
vXvk−1
v0
O1
m−1
v0
O1
n−1
k−1
Xv |Δλv |
v0
k
k
vk pn−v−1
qv1
|sv |k |Δλv |
m vk qk |s |k
v1 v
v0
O1
k
k
vk pn−v−1
qv1
|sv |k |Δλv |Xv1−k
|Δλv |
∞ pk
n−v−1
nv1
nPnk
v|Δλv |
Δv|Δλv |
v
qrk |sr |k
r0
rXrk−1
m−1
m−1
v0
v0
|Δλv | Xv O1
m|Δλm |
m
qvk |sv |k
v0
vXvk−1
vΔ2 λv Xv O1|Δλm | Xm O1,
8
Journal of Inequalities and Applications
m
1
k
n1 nPn
|Tn5 |k m
k
1 np0 qn sn λn k
n1 nPn
O1
m
nk−1 Pn−k qnk |sn |k |λn |k
n1
O1
m
n−1 qnk |sn |k |λn |k
n1
O1,
as in the case of Tn1 .
3.4
This completes the proof of the theorem.
References
1 G. Das, “On some methods of summability,” The Quarterly Journal of Mathematics Oxford Series, vol. 17,
no. 2, pp. 244–256, 1966.
2 W. T. Sulaiman, “Notes on two summability methods,” Pure and Applied Mathematika Sciences, vol. 31,
no. 1-2, pp. 59–69, 1990.
3 W. T. Sulaiman, “Extension on absolute summability factors of infinite series,” Journal of Mathematical
Analysis and Applications, vol. 322, no. 2, pp. 1224–1230, 2006.
4 N. Singh and N. Sharma, “On N, p, q summability factors of infinite series,” Proceedings of
Mathematical Sciences, vol. 110, no. 1, pp. 61–68, 2000.