We establish a summability factor theorem for summability|A, δ| k, whereA is lower triangular matrix with nonnegative entries satisfying certain conditions.. It should be noted that ever
Trang 1Volume 2010, Article ID 105136, 10 pages
doi:10.1155/2010/105136
Research Article
A Summability Factor Theorem for
Quasi-Power-Increasing Sequences
E Savas¸
Department of Mathematics, ˙Istanbul Ticaret University, ¨ Usk ¨udar, 34378 ˙Istanbul, Turkey
Correspondence should be addressed to E Savas¸,ekremsavas@yahoo.com
Received 23 June 2010; Revised 3 September 2010; Accepted 15 September 2010
Academic Editor: J Szabados
Copyrightq 2010 E Savas¸ 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
We establish a summability factor theorem for summability|A, δ| k, whereA is lower triangular
matrix with nonnegative entries satisfying certain conditions This paper is an extension of the main result of the work by Rhoades and Savas¸2006 by using quasi f-increasing sequences.
1 Introduction
Recently, Rhoades and Savas¸1 obtained sufficient conditions fora n λ nto be summable
|A, δ| k,k ≥ 1 by using almost increasing sequence The purpose of this paper is to obtain the
corresponding result for quasif-increasing sequence.
A sequence{λ n} is said to be of bounded variation bv if n |Δλ n | < ∞ Let bv0
bv∩ c0 , where c0denotes the set of all null sequences
LetA be a lower triangular matrix, {s n} a sequence Then
A n:n
ν0
a nν s ν 1.1
A series
a n, with partial sumss n , is said to be summable |A| k , k ≥ 1 if
∞
n1
n k−1 |A n − A n−1|k < ∞, 1.2
and it is said to be summable|A, δ| k , k ≥ 1 and δ ≥ 0 if see, 2
∞
n1
n δk k−1 |A n − A n−1|k < ∞. 1.3
Trang 2A positive sequence{b n} is said to be an almost increasing sequence if there exist an increasing sequence{c n } and positive constants A and B such that Ac n ≤ b n ≤ Bc nsee, 3 Obviously, every increasing sequence is almost increasing However, the converse need not
be true as can be seen by taking the example, sayb n e−1n
n.
A positive sequenceγ : {γ n } is said to be a quasi β-power increasing sequence if there
exists a constantK Kβ, γ ≥ 1 such that
Kn β γ n ≥ m β γ m 1.4
holds for alln ≥ 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, sayγ n n −βforβ > 0 see, 4 A sequence satisfying 1.4 for
β 0 is called a quasi-increasing sequence It is clear that if {γ n } is quasi β-power increasing
then{n β γ n} is quasi-increasing
A positive sequenceγ {γ n } is said to be a quasi-f-power increasing sequence if there
exists a constantK Kγ, f ≥ 1 such that Kf n γ n ≥ f m γ m holds for alln ≥ m ≥ 1, where
f : {f n } {n β log n μ }, μ > 0, 0 < β < 1, see, 5
We may associate withA two lower triangular matrices A and A as follows:
a nv n
rv a nr , n, v 0, 1, ,
a nv a nv − a n−1,v , n 1, 2, ,
1.5
where
a00 a00 a00 1.6
Given any sequence{x n }, the notation x n O1 means that x n O1 and 1/x n
O1 For any matrix entry a nv , Δ v a nv : anv − a n,v 1
Rhoades and Savas¸1 proved the following theorem for |A, δ| ksummability factors
of infinite series
Theorem 1.1 Let {X n } be an almost increasing sequence and let {β n } and {λ n } be sequences such
that
i |Δλ n | ≤ β n ,
ii lim β n 0,
iii∞n1 n|Δβ n |X n < ∞,
iv |λ n |X n O1.
Let A be a lower triangular matrix with nonnegative entries satisfying
v na nn O1,
vi a n−1,ν ≥ a nν for n ≥ ν 1,
vii a n0 1 for all n,
Trang 3viiin−1 ν1 a νν a nν 1 Oa nn ,
ixm 1 nν 1 n δk|Δν a nν | Oν δk a νν and
xm 1 nν 1 n δk a nν 1 Oν δk .
If
xim
n1 n δk−1 |t n|k OX m , where t n: 1/n 1n
k1 ka k , then the series
a n λ n is summable |A, δ| k , k ≥ 1.
It should be noted that, if{X n} is an almost increasing sequence, then condition iv implies that the sequence{λ n } is bounded However, if {X n } is a quasi β-power increasing
sequence or a quasi f-increasing sequence, iv does not imply that λ is bounded For
example, the sequence{X m } defined by X m m −β is trivially a quasiβ-power increasing
sequence for eachβ > 0 If λ {m δ }, for any 0 < δ < β, then λ m X m m δ−β O1, but λ is not
bounded,see, 6,7
The purpose of this paper is to prove a theorem by using quasif-increasing sequences.
We show that the crucial condition of our proof,{λ n} ∈ bv0, can be deduced from another
condition of the theorem
2 The Main Results
We now will prove the following theorems
Theorem 2.1 Let A satisfy conditions (v)–(x) and let {β n } and {λ n } be sequences satisfying
conditions (i) and (ii) of Theorem 1.1 and
m
n1
λ n om, m −→ ∞. 2.1
If {X n } is a quasi f-increasing sequence and condition (xi) and
∞
n1
nX n
β, μ Δβ n < ∞ 2.2
are satisfied then the series
a n λ n is summable |A, δ| k , k ≥ 1, where {f n } : {n β log n μ }, μ ≥
0, 0 ≤ β < 1, and X n β, μ : n β log n μ X n .
The following theorem is the special case ofTheorem 2.1forμ 0.
Theorem 2.2 Let A satisfy conditions (v)–(x) and let {β n } and {λ n } be sequences satisfying
conditions (i), (ii), and2.1 If {X n } is a quasi β-power increasing sequence for some 0 ≤ β < 1
and conditions (xi) and
∞
n1
nX nβ Δβ n < ∞ 2.3
are satisfied, where X n β : n β X n , then the seriesa n λ n is summable |A, δ| k , k ≥ 1.
Trang 4Remark 2.3 The conditions {λ n} ∈ bv0, and iv do not appear among the conditions of
Theorems2.1and2.2 ByLemma 3.3, under the conditions on{X n }, {β n }, and {λ n} as taken
in the statement of theTheorem 2.1, also in the statement ofTheorem 2.2with the special case
μ 0, conditions {λ n} ∈ bv0andiv hold
3 Lemmas
We will need the following lemmas for the proof of our mainTheorem 2.1
Lemma 3.1 see 8 Let {ϕ n } be a sequence of real numbers and denote
Φn:n
k1
ϕ k , Ψn:∞
kn
Δϕ k . 3.1
IfΦn on then there exists a natural number N such that
ϕ n ≤ 2Ψn 3.2
for all n ≥ N.
Lemma 3.2 see 9 If {X n } is a quasi f-increasing sequence, where {f n } {n β log n μ }, μ ≥
0, 0 ≤ β < 1, then conditions 2.1 of Theorem 2.1 ,
m
n1
|Δλ n | om, m −→ ∞, 3.3
∞
n1
nX nβ, μ|Δ|Δλ n || < ∞, 3.4
where X n β, μ n β log n μ X n , imply conditions (iv) and
λ n −→ 0, n −→ ∞. 3.5
Lemma 3.3 see 7 If {X n } is a quasi f-increasing sequence, where {f n } {n β log n μ }, μ ≥
0, 0 ≤ β < 1, then under conditions (i), (ii), 2.1, and 2.2, conditions (iv) and 3.5 are satisfied.
Lemma 3.4 see 7 Let {X n } be a quasi f-increasing sequence, where {f n } {n β log n μ }, μ ≥
0, 0 ≤ β < 1 If conditions (i), (ii), and 2.2 are satisfied, then
nβ n X n O1, 3.6
∞
n1
Trang 54 Proof of Theorem 2.1
Proof Let y n be the nth term of the A transform of the partial sums ofn
i0 λ i a i Then we have
y n:n
i0
a ni s in
i0
a nii ν0
λ ν a ν
n
ν0
λ ν a νn iν
a nin
ν0
a nν λ ν a ν ,
4.1
and, forn ≥ 1, we have
Y n: yn − y n−1n
ν0
a nν λ ν a ν 4.2
We may writenoting that vii implies that a n0 0,
Y nn
ν1
a
nν λ ν ν
νa ν
n
ν1
a
nν λ ν ν
ν
r1
ra r−ν−1
r1
ra r
n−1
ν1
Δν
a
nν λ ν ν
ν
r1
ra r a nn λ n
n
n
r1
ra r
n−1
ν1
Δν a nν λ ν ν 1 ν t ν n−1
ν1
a n,ν 1 Δλ νν 1 ν t ν
n−1
ν1
a n,ν 1 λ ν 11
ν t ν
n 1a nn λ n t n n
T n1 T n2 T n3 T n4 , say.
4.3
To complete the proof it is sufficient, by Minkowski’s inequality, to show that
∞
n1
n δk k−1 |T nr|k < ∞, for r 1, 2, 3, 4. 4.4
From the definition of A and using vi and vii it follows that
Trang 6Using H ¨older’s inequality
I1:m
n1
n δk k−1 |T n1|km
n1
n δk k−1
n−1
ν1
Δν a nν λ ν ν 1 ν t ν
k
O1 m 1
n1
n δk k−1
n−1
ν1
|Δν a nν ||λ ν ||t ν|
k
O1 m 1
n1
n δk k−1
n−1
ν1
|Δν a nν ||λ ν|k |t ν|k n−1
ν1
|Δν a nν|
k−1 ,
Δν a nν a nν − a n,ν 1
a nν − a n−1,ν − a n,ν 1 a n−1,ν 1
a nν − a n−1,ν ≤ 0.
4.6
Thus, usingvii,
n−1
ν0
|Δν a nν| n−1
ν0
a n−1,ν − a nν 1 − 1 a nn a nn 4.7 Sinceλ n is bounded byLemma 3.3, usingv, ix, xi, i, and condition 3.7 ofLemma 3.4
I1 O1 m 1
n1
n δk na nnk−1n−1
ν1
|λ ν|k |t ν|k|Δν a nν|
O1 m 1
n1
n δk
n−1
ν1
|λ ν|k−1 |λ ν||Δν a nν ||t ν|k
O1m
ν1
|λ ν ||t ν|k m 1
nν 1
n δk|Δν a nν|
O1m
ν1
ν δk |λ ν |a νν |t ν|k
O1m
ν1
ν δk−1 |λ ν ||t ν|k
O1 m−1
ν1 Δ|λ ν|ν
r1
r δk−1 |t r|k |λ m|m
r1
r δk−1 |t r|k
O1 m−1
ν1
|Δλ ν |X ν O1|λ m |X m
O1m
ν1
β ν X ν O1|λ m |X m
O1.
4.8
Trang 7Using H ¨older’s inequality,
I2:m 1
n2
n δk k−1 |T n2|km 1
n2
n δk k−1
n−1
ν1
a n,ν 1 Δλ νν 1 ν t ν
k
O1 m 1
n2
n δk k−1 n−1
ν1
a n,ν 1 |Δλ ν ||t ν|
k
O1 m 1
n2
n δk k−1 n−1
ν1
|Δλ ν ||t ν|k a n,ν 1 n−1
ν1
a n,ν 1 |Δλ ν|
k−1
.
4.9
ByLemma 3.1, condition3.3, in view ofLemma 3.3implies that
∞
n1
|Δλ n| ≤ 2∞
n1
∞
kn
|Δ|Δλ k|| 2∞
k1
|Δ|Δλ k|| 4.10
holds Thus byLemma 3.3,3.4 implies that∞n1 |Δλ n| converges Therefore, there exists a positive constantM such that∞
n1 |Δλ n | ≤ M and from the properties of matrix A, we obtain
n−1
ν1
a n,ν 1 |Δλ k | ≤ Ma nn 4.11
We have, usingv and x,
I2 O1 m 1
n2
n δk na nnk−1 n−1
ν1
a n,ν 1 β ν |t ν|k
O1m
ν1
β ν |t ν|k m 1
nν 1
n δk a n,ν 1
4.12
Therefore,
I2 O1m
ν1
ν δk β ν |t ν|k
O1m
ν1
νβ ν |t ν|k
ν ν δk .
4.13
Trang 8Using summation by parts,2.2, xi, and condition 3.6 and 3.7 ofLemma 3.4
I2: O1 m−1
ν1
Δνβ νν r1
r δk−1 |t r|k O1mβ m
m
r1
r δk−1 |t r|k
O1 m−1
ν1
ν Δβ ν X ν O1 m−1
ν1
β ν 1 X ν 1 O1
O1.
4.14
Using H ¨older’s inequality andviii,
m 1
n2
n k−1 |T n3|km 1
n2
n δk k−1
n−1
ν1
a n,ν 1 λ ν 11
ν t ν
k
≤m 1
n2
n δk k−1 n−1
ν1
|λ ν 1|a n,ν 1 ν |t ν|
k
O1 m 1
n2
n δk k−1 n−1
ν1
|λ ν 1 |a n,ν 1 |t ν |a νν
k
O1 m 1
n2
n δk k−1 n−1
ν1
|λ ν 1|k a νν |t ν|k a n,ν 1
n−1
ν1
a νν |a n,ν 1|
k−1
.
4.15
Using boundedness of{λ n}, v, x, xi, Lemmas3.3and3.4
I3 O1 m 1
n2
n δk na nnk−1n−1
ν1
|λ ν 1|k a νν |t ν|k a n,ν 1
O1m
ν1
|λ ν 1 |a νν |t ν|k m 1
nν 1
n δk a n,ν 1
O1m
ν1
|λ ν 1 |ν δk a νν |t ν|k
O1m
v1
|λ v 1 |va vv v δk−1 |t v|k
O1m
v1
|λ v 1 |v δk−1 |t v|k
4.16
Trang 9Using summation by parts
I3 O1 m−1
v1
|Δλ v 1|v
r1
r δk−1 |t r|k O1|λ m 1|m
v1
v δk−1 |t v|k
O1 m−1
v1
|Δλ v 1|v 1
r1
r δk−1 |t r|k O1|λ m 1|m 1
v1
v δk−1 |t v|k
O1 m−1
v1
|Δλ v 1 |X v 1 O1|λ m 1 |X m 1
O1 m−1
v1
β v 1 X v 1 O1|λ m 1 |X m 1
O1.
4.17
Finally, using boundedness of{λ n}, and v we have
m
n1
n δk k−1 |T n4|km
n1
n δk k−1
n 1a nn λ n t n n
k
O1m
n1
n δk a nn |λ n ||t n|k
O1,
4.18
as in the proof ofI1.
5 Corollaries and Applications to Weighted Means
Setting δ 0 in Theorem 2.1 and Theorem 2.2 yields the following two corollaries, respectively
Corollary 5.1 Let A satisfy conditions (v)–(viii) and let {β n } and {λ n } be sequences satisfying
conditions (i), (ii), and 2.1 If {X n } is a quasi f-increasing sequence, where {f n} :
{n β log n μ }, μ ≥ 0, 0 ≤ β < 1, and conditions 2.2 and
m
n1
1
n |t n|k OX m , m −→ ∞, 5.1
are satisfied then the series
a n λ n is summable |A| k , k ≥ 1.
Proof If we take δ 0 inTheorem 2.1then conditionxi reduces condition 5.1
Corollary 5.2 Let A satisfy conditions (v)–(viii) and let {β n } and {λ n } be sequences satisfying
conditions (i), (ii), and2.1 If {X n } is a quasi β-power increasing sequence for some 0 ≤ β < 1 and
conditions2.3 and 5.1 are satisfied then the seriesa n λ n is summable |A| k , k ≥ 1.
Trang 10Corollary 5.3 Let {p n } be a positive sequence such that P n:n i0 p i → ∞, as n → ∞ satisfies
np n OP n , as n −→ ∞, 5.2
m 1
nv 1
n δk p n
P n P n−1 O
v δk
and let {β n } and {λ n } be sequences satisfying conditions (i), (ii), and 2.1 If {X n } is a quasi
f-increasing sequence, where {f n } : {n β log n μ }, μ ≥ 0, 0 ≤ β < 1, and conditions (xi) and 2.2 are
satisfied then the series,a
n λ n is summable |N, p n , δ| k for k ≥ 1.
Proof In Theorem 2.1, set A N, p n Conditions i and ii of Corollary 5.3 are, respectively, conditionsi and ii ofTheorem 2.1 Conditionv becomes condition 5.2 and conditionsix and x become condition 5.3 for weighted mean method Conditions vi,
vii, and viii ofTheorem 2.1are automatically satisfied for any weighted mean method The following Corollary is the special case ofCorollary 5.3forμ 0.
Corollary 5.4 Let {p n } be a positive sequence satisfying 5.2, 5.3 and let {X n } be a quasi β-power
increasing sequence for some 0 ≤ β < 1 Then under conditions (i), (ii), (xi), 2.1, and 2.3,a n λ n
is summable |N, p n , δ| k , k ≥ 1.
References
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... is summable |A| k , k ≥ 1. Trang 10Corollary 5.3 Let {p n... class="text_page_counter">Trang 8
Using summation by parts,2.2, xi, and condition 3.6 and 3.7 ofLemma 3.4
I2:...
4.16
Trang 9Using summation by parts
I3 O1 m−1