Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 814974, 10 pages
doi:10.1155/2010/814974
Research Article
A Note on Generalized |A|k -Summability Factors
for Infinite Series
Ekrem Savaş
Department of Mathematics, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey
Correspondence should be addressed to Ekrem Savaş, ekremsavas@yahoo.com
Received 9 September 2009; Revised 9 November 2009; Accepted 14 December 2009
Academic Editor: Martin Bohner
Copyright q 2010 Ekrem Savaş. 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.
A general theorem concerning the |A|k —summability factors of infinite series has been proved.
1. Introduction
A weighted mean matrix, denoted by N, pn , is a lower triangular matrix with entries pk /Pn ,
where {pk } is a nonnegative sequence with p0 > 0, and Pn : nk0 pk .
Mishra and Srivastava 1 obtained sufficient conditions on a sequence {pk } and a
sequence {λn } for the series an Pn λn /npn to be absolutely summable by the weighted mean
matrix N, pn .
Recently Savaş and Rhoades 2 established the corresponding result for a nonnegative
triangle, using the correct definition of absolute summability of order k ≥ 1.
Let A be an infinite lower triangular matrix. We may associate with A two lower
whose entries are defined by
triangular matrices A and A,
ank n
ani ,
ank ank − an−1,k ,
ik
1.1
respectively. The motivation for these definitions will become clear as we proceed.
Let A be an infinite matrix. The series ak is said to be absolutely summable by A, of
order k ≥ 1, written as |A|k , if
∞
nk−1 |Δtn−1 |k < ∞,
k0
1.2
2
Journal of Inequalities and Applications
where Δ is the forward difference operator and tn denotes the nth term of the matrix
transform of the sequence {sn }, where sn : nk1 ak .
Thus
tn tn − tn−1 n
ank sk n
n
n
k
n
ank aν aν ank anν aν ,
k1
k1
n
n−1
anν aν −
ν1
ν1
ν1
an−1,ν aν ν1
n
kν
ν1
1.3
anν aν ,
ν1
since an−1,n 0.
A sequence {λn } is said to be of bounded variation bv if n |Δ λn | < ∞. Let bv0 bv ∩ c0 , where c0 denotes the set of all null sequences.
A positive sequence {bn } is said to be an almost increasing sequence if there exist an
increasing sequence {cn } and positive constants A and B such that Acn ≤ bn ≤ Bcn , see 3.
Obviously, every increasing sequence is almost increasing. However, the converse need not
n
be true as can be seen by taking the example, say bn e−1 n.
A positive sequence γ : {γn } is said to be a quasi β-power increasing sequence if there
exists a constant K Kβ, γ ≥ 1 such that
Knβ γn ≥ mβ γm
1.4
holds for all n ≥ m ≥ 1. It should be noted that every almost increasing sequence is quasi
β-power increasing sequence for any nonnegative β, but the converse need not be true as can
be seen by taking an example, say γn n−β for β > 0 see 4. If 1.4 stays with β 0 then γ
is simply called a quasi-increasing sequence. It is clear that if {γn } is quasi β-power increasing
then {nβ γn } is quasi-increasing.
A positive sequence γ {γn } is said to be a quasi-f-power increasing sequence, if
there exists a constant K Kγ, f ≥ 1 such that Kfn γn ≥ fm γm holds for all n ≥ m ≥ 1, where
f : {fn } {nβ log nμ }, μ > 0, 0 < β < 1 was considered instead of nβ see 5, 6.
Given any sequence {xn }, the notation xn O1 means xn O1 and 1/xn O1.
Quite recently, Savaş and Rhoades 2 proved the following theorem for |A|k summability factors of infinite series.
Theorem 1.1. Let A be a triangle with nonnegative entries satisfying
i an0 1, n 0, 1, . . . ,
ii an−1,ν ≥ anν for n ≥ ν 1,
iii nann O1,
iv Δ1/ann O1, and
v nν0 aνν |
an,ν
1 | Oann .
If {Xn } is a positive nondecreasing sequence and the sequences {λn } and {βn } satisfy
vi |Δλn | ≤ βn ,
vii lim βn 0,
viii |λn |Xn O1,
Journal of Inequalities and Applications
ix
∞
n1
x Tn :
then the series
3
nXn |Δβn | < ∞, and
n
ν1
∞
n1
|sν |k
OXn ,
ν
an λn /nann is summable |A|k , k ≥ 1.
It should be noted that if {Xn } is an almost increasing sequence then viii implies that
the sequence {λn } is bounded. However, when {Xn } is a quasi β-power increasing sequence
or a quasi f-increasing sequence, viii does not imply |λm | O1, m → ∞. For example,
since Xm m−β is a quasi β-power increasing sequence for 0 < β < 1, if we take λm mδ ,
O1 see 7.
0 < δ < β < 1 then |λm |Xm mδ−β O1, m → ∞ holds but |λm | mδ /
The goal of this paper is to prove a theorem by using quasi f-increasing sequences.
We show that the crucial condition of our proof, {λn } ∈ bv0 , can be deduced from another
condition of the theorem.
2. The Main Results
We have the following theorem:
Theorem 2.1. Let A be nonnegative triangular matrix satisfying conditions i–v and let
{βn } and {λn } be sequences satisfying conditions (vi) and (vii) of Theorem 1.1 and
m
λn om,
m −→ ∞.
2.1
n1
If {Xn } is a quasi f-increasing sequence and condition (x) and
∞
nXn β, μ Δβn < ∞
2.2
n1
μ
β
are satisfied, then the series ∞
n1 an λn /nann is summable |A|k , k ≥ 1, where {fn } : {n log n },
μ
μ ≥ 0, 0 ≤ β < 1, and Xn β, μ : nβ log n Xn .
Theorem 2.1 includes the following theorem with the special case μ 0.
Theorem 2.2. Let A satisfying conditions i–v and let {βn } and {λn } be sequences satisfying
conditions (vi), (vii), and 2.1. If {Xn } is a quasi β-power increasing sequence for some 0 ≤ β < 1
and conditions (x) and
∞
nXn β Δβn < ∞
2.3
n1
are satisfied, where Xn β : nβ Xn , then the series
∞
ν1
an λn /nann is summable |A|k , k ≥ 1.
If we take that {Xn } is an almost increasing sequence instead of a quasi β-power
increasing sequence then our Theorem 2.2 reduces to 8, Theorem 1.
4
Journal of Inequalities and Applications
Remark 2.3. The crucial condition, {λn } ∈ bv0 , and condition viii do not appear among the
conditions of Theorems 2.1 and 2.2. By Lemma 3.3, under the conditions on {Xn }, {βn }, and
{λn } as taken in the statement of the Theorem 2.1, also in the statement of Theorem 2.2 with
the special case μ 0, conditions {λn } ∈ bv0 and viii hold.
3. Lemmas
We shall need the following lemmas for the proof of our main Theorem 2.1.
Lemma 3.1 see 9. Let {ϕn } be a sequence of real numbers and denote
Φn :
n
ϕk ,
Ψn :
k1
∞ Δϕk .
kn
3.1
If Φn on then there exists a natural number N such that
ϕn ≤ 2Ψn
3.2
for all n ≥ N.
Lemma 3.2 see 7. If {Xn } is a quasi f-increasing sequence, where {fn } {nβ log nμ }, μ ≥ 0,
0 ≤ β < 1, then conditions 2.1 of Theorem 2.1,
m
|Δλn | om,
m −→ ∞,
3.3
n1
∞
nXn β, μ |Δ|Δλn || < ∞,
3.4
n1
where Xn β, μ nβ log nμ Xn , imply conditions viii and
λn −→ 0,
n −→ ∞.
3.5
Lemma 3.3 see 7. If {Xn } is a quasi f-increasing sequence, where {fn } {nβ log nμ }, μ ≥ 0,
0 ≤ β < 1, then under conditions (vi), (vii), 2.1 and 2.2, conditions viii and 3.5 are satisfied.
Lemma 3.4 see 7. Let {Xn } be a quasi f-increasing sequence, where {fn } {nβ log nμ }, μ ≥ 0,
0 ≤ β < 1. If conditions (vi), (vii), and 2.2 are satisfied, then
nβn Xn O1,
∞
n1
βn Xn < ∞.
3.6
3.7
Journal of Inequalities and Applications
5
4. Proof of Theorem 2.1
Let Tn denote the nth term of the A-transform of the partial sums of the series
∞
n1 an λn /nann . Then, we have
Tn n
anν
ν1
ν
ai λi
i1
aii i
m
n
n
ai λi ai λi
ani
.
anν a
i
aii i
ii
νi
i1
i1
4.1
Thus,
Tn − Tn−1 n
n−1
ai λi ai λi
− an−1,i
ani
a
i
aii i
ii
i1
i1
n
n
ai λi ai λi
ani
ani − an−1,i aii i
aii i
i1
i1
n
λi
ani
si − si−1 a
ii i
i1
n−1
n
λi
λn
λi si−1
si ann
sn − ani
ani
aii i
ann n
aii i
i1
i1
4.2
n−1
n−1
λi
λn
λi
1 si
si ann
sn − an,i
1
ani
a
i
a
n
1ai
1,i
1
i
ii
nn
i1
i1
n−1 λi
λi
1
λn
− an,i
1
.
ani
si ann
a
i
na
1a
i
ii
i
1,i
1
nn
i1
It is easy to see that
an,i
1 λi
1
an,i
1
ani λi
ani
−
Δi
Δλi .
λi iaii
iaii
i 1ai
1,i
1
i 1ai
1,i
1
4.3
Also we may write
Δi
ani λi
Δi 1
ani
1
an,i
1 λi
−
λi .
iaii
iaii
iaii i 1ai
1,i
1
4.4
Therefore, for n > 1,
Tn − Tn−1 n−1
Δi ani i1
iaii
λi si n−1
an,i
1 λi
i1
n−1
1
1
−
si
iaii i 1ai
1,i
1
an,i
1
λn
Δi λi si sn
n
1a
i
i
1,i
1
i1
Tn1 Tn2 Tn3 Tn4 ,
say.
4.5
6
Journal of Inequalities and Applications
To complete the proof of the theorem, it will be sufficient to show that
∞
nk−1 |Tnr |k < ∞,
for r 1, 2, 3, 4.
4.6
n1
Using Hölder’s inequality and condition iii,
I1 m
1
k−1
n
k
|Tn1 | ≤
n1
m
1
k−1
n
n1
O1
m
1
k−1
n
ii
i1
k
n−1
ani λi si |
|Δi n1
O1
k
n−1 Δ
a
i
ni
ia λi si 4.7
i1
m
1
k−1
n
k−1
n−1
n−1
k
k
×
.
ani ||λi | |si |
ani |
|Δi |Δi n1
i1
i1
Since λn is bounded by Lemma 3.3, using ii, iii, vi, x, and property 3.7 of
Lemma 3.4,
I1 O1
m
1
n−1
ani |
|λi |k |si |k |Δi n1
i1
nann k−1
m
1
n−1
k−1
k
k−1
O1 nann ani ||si |
|λi | |λi ||Δi n1
O1
m
i1
i1
O1
m
m
1
|λi ||si |k
ani |
nann k−1 |Δi ni
1
|λi ||si |k aii O1
m
|λi ||si |k
i1
O1
m
i1
O1
O1
i1
|λi |
i
|sr |k
r1
r
m−1
i
1
i1
r1
Δ|λi |
r
−
i
m−1
i
|sr |k
i0
r1
|λi
1 |
|sr |k O1|λm |
m−1
Δ|λi |Xi O1|λm |Xm
i1
O1
m
i1
βi Xi O1|λm |Xm O1.
4.8
r
m
|si |k
i1
i
Journal of Inequalities and Applications
7
Now
I2 m
1
k−1
n
k
|Tn2 | n1
m
1
n
n1
O1
m
1
n−1
n1
i1
nk−1
k
n−1
1
si an,i
1 λi Δ
i1
iaii
k−1 k
1 an,i
1 ||λi |Δ
|si | .
|
ia 4.9
ii
From 2,
1
1
1
1
Δ
Δ
.
iaii
aii
iaii
i 1
4.10
Thus, using iv and ii,
Δ 1 1 Δ 1 1 iaii
i
1
aii
iaii 4.11
1
O1 O1.
i
1
Hence, using Hölder’s inequality, v, iii, and the fact that the λn ’s are bounded,
I2 O1
m
1
k−1
n
n1
O1
m
1
k−1
n
n1
O1
O1
n−1
an,i
1 |aii |λi ||si |
|
m
1
k−1
n
n−1
k
an,i
1 |aii |λi | |si |
|
k
n−1
i1
m
1
n−1
an,i
1 |aii |λi |k |si |k
|
n1
i1
nann k−1
m
|λi |k |si |k aii
m
m
i1
m
1
an,i
1 |
nann k−1 |
ni
1
|λi |k |si |k aii
i1
O1
k
i1
i1
O1
k
i1
n1
O1
n−1
1
an,i
1 ||λi |
|
|si |
i
1
i1
m
1
an,i
1 |
|
ni
1
|λi |k |si |k aii
k−1
aii |
an,i
1 |
8
Journal of Inequalities and Applications
O1
m
|λi ||λi |k−1 |si |k
i1
1
i
m
|si |k
O1,
|λi |
i
i1
4.12
as in the proof of I1 .
It follows from 3.6 that βn O1/n and hence that |Δλn | O1/n by condition
vi.
Using iii, Hölder’s inequality, and v,
I3 m
1
k−1
n
k
|Tn3 | n1
O1
m
1
k−1
n
m
1
k−1
n
m
1
k−1
n
m
1
O1
O1
k
an,i
1 ||Δλi ||si |
|
n−1
an,i
1 |
|
aii
|Δλi |k |si |k
k
aii
i1
n−1
n−1
an,i
1 |
|
aii
|Δλi |k |si |k
k
a
i1
ii
n−1
m
1
1
n1 i1
akii
an,i
1 ||Δλi |k |si |k
|
k−1
aii |
an,i
1 |
i1
4.13
aii
m
aii
m
1
k
k
an,i
1 |
|Δλ
|
|s
|
|
i
i
k
i1 aii
ni
1
m |Δλi | k−1
i0
O1
aii
nann k−1
n1
O1
an,i
1 ||Δλi ||si |
|
n−1
aii
i1
n1
O1
k
i1
n1
O1
n
n−1
k
n−1
an,i
1 Δλi si i1 i 1ai
1,i
1 k−1 n1
n1
O1
m
1
aii
|Δλi ||si |k
m
m
|Δλi | |si |k O1 |si |k βi .
i0
i1
Since |si |k iTi − Ti−1 by x, we have
I3 O1
m
i1
iTi − Ti−1 βi .
4.14
Journal of Inequalities and Applications
9
Using Abel’s transformation, vi, 2.2, and properties 3.7 and 3.6 of Lemma 3.4,
I3 O1
m−1
Ti Δ iβi O1mTn βn
i1
O1
m−1
m−1
iΔβi Xi O1 Xi βi O1mXn βn O1.
i1
4.15
i1
Using the boundedness of λn and x,
I4 m
1
nk−1 |Tn4 |k n1
sn λn k
nk−1 n n1
m
1
m
1
|sn |k
1 m
1
|sn |k |λn |k |λn ||λn |k−1 O1,
n n1 n
n1
4.16
as in the proof of I1 .
A weighted mean matrix, written N, pn , is a lower triangular matrix with entries
anv pv /Pn , where {pn } is a nonnegative sequence with p0 > 0 and Pn : ni0 pi → ∞, as
n → ∞.
Corollary 4.1. Let {pn }be a positive sequence satisfying
i npn OPn and
ii ΔPn /pn O1.
and let {βn } and {λn } be sequences satisfying conditions vi, vii, and 2.1. If {Xn } is a quasi
f-increasing sequence, where {fn } : {nβ log nμ }, μ ≥ 0, 0 ≤ β < 1, and conditions (x) and 2.2
are satisfied, then the series ∞
n1 an Pn λn /npn is summable |N, pn |k , k ≥ 1.
Acknowledgment
The author wishes to thank the referees for their careful reading of the manuscript and for
their helpful suggestions.
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