Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 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¸,
Received 23 June 2010; Revised 3 September 2010; Accepted 15 September 2010
Academic Editor: J. Szabados
Copyright q 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 for
n
ν0
a
nν
s
ν
. 1.1
A series
a
n
, with partial sums s
n
, is said to be summable |A|
k
,k ≥ 1if
∞
n1
n
k−1
|
A
n
− A
n−1
|
k
< ∞,
see, 3.
Obviously, every increasing sequence is almost increasing. However, the converse need not
be true as can be seen by taking the example, say b
n
e
−1
n
n.
A positive sequence γ : {γ
n
} is said to be a quasi β-power increasing sequence if there
exists a constant K Kβ, γ ≥ 1 such that
Kn
β
γ
n
≥ m
β
γ
m
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 γ
n
n
−β
for β>0 see, 4. A sequence satisfying 1.4 for
β 0 is called a quasi-increasing sequence. It is clear that if {γ
n
nv
n
rv
a
nr
,n,v 0, 1, ,
a
nv
a
nv
− a
n−1,v
,n 1, 2, ,
1.5
where
a
00
a
00
a
00
. 1.6
Given any sequence {x
n
}, the notation x
n
O1 means that x
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
m1
nν1
n
δk
a
nν1
Oν
δk
.
If
xi
m
n1
n
δk−1
|t
n
|
k
OX
m
,wheret
n
:1/n 1
n
k1
ka
k
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 quasi f-increasing sequences.
We show that the crucial condition of our proof, {λ
n
}∈bv
0
, 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
o
m
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 of Theorem 2.1 for μ 0.
Theorem 2.2. Let A satisfy conditions (v)–(x) and let {β
n
} and {λ
n
} be sequences satisfying
conditions (i), (ii), and 2.1.If{X
n
} is a quasi β-power increasing sequence for some 0 ≤ β<1
and conditions (xi) and
∞
n1
nX
n
β
n
}, {β
n
},and{λ
n
} 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 {λ
n
}∈bv
0
and iv hold.
3. Lemmas
We will need the following lemmas for the proof of our main Theorem 2.1.
Lemma 3.1 see 8. Let {ϕ
n
} be a sequence of real numbers and denote
Φ
n
:
n
k1
ϕ
k
, Ψ
n
:
∞
},μ ≥
0, 0 ≤ β<1, then conditions 2.1 of Theorem 2.1,
m
n1
|
Δλ
n
|
o
m
,m−→ ∞ ,
3.3
∞
n1
nX
n
β, μ
|
Δ
|
Δλ
n
||
< ∞,
μ
}, μ ≥
0, 0 ≤ β<1. If conditions (i), (ii), and 2.2 are satisfied, then
nβ
n
X
n
O
1
, 3.6
∞
n1
β
n
X
n
< ∞.
3.7
Journal of Inequalities and Applications 5
4. Proof of Theorem 2.1
Proof. Let y
n
be the nth term of the A transform of the partial sums of
n
i0
λ
ν0
λ
ν
a
ν
n
iν
a
ni
n
ν0
a
nν
λ
ν
a
ν
,
4.1
and, for n ≥ 1, we have
Y
n
: y
n
− y
n−1
n
ν1
a
nν
λ
ν
ν
ν
r1
ra
r
−
ν−1
r1
ra
r
n−1
ν1
Δ
ν
a
nν
λ
ν
ν 1
ν
t
ν
n−1
ν1
a
n,ν1
Δλ
ν
ν 1
ν
t
ν
n−1
ν1
a
n,ν1
λ
n
δkk−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
a
n,ν1
≥ 0. 4.5
6 Journal of Inequalities and Applications
Using H
¨
older’s inequality
I
1
:
m
n1
n
δkk−1
|
T
n1
k
O
1
m1
n1
n
δkk−1
n−1
ν1
|
Δ
ν
a
nν
||
λ
ν
||
t
ν
|
k
O
ν1
|
Δ
ν
a
nν
|
k−1
,
Δ
ν
a
nν
a
nν
− a
n,ν1
a
nν
− a
n−1,ν
− a
n,ν1
a
n−1,ν1
a
nν
− a
n−1,ν
is bounded by Lemma 3.3,usingv, ix, xi, i, and condition 3.7 of Lemma 3.4
I
1
O
1
m1
n1
n
δk
na
nn
k−1
n−1
ν1
|
λ
ν
|
k
|
t
ν
|
k
a
nν
||
t
ν
|
k
O
1
m
ν1
|
λ
ν
||
t
ν
|
k
m1
nν1
n
δk
|
Δ
ν1
ν
δk−1
|
λ
ν
||
t
ν
|
k
O
1
m−1
ν1
Δ
|
λ
ν
|
ν
r1
r
Δλ
ν
|
X
ν
O
1
|
λ
m
|
X
m
O
1
m
ν1
β
ν
X
ν
O
1
m1
n2
n
δkk−1
n−1
ν1
a
n,ν1
Δλ
ν
ν 1
ν
t
ν
k
O
n
δkk−1
n−1
ν1
|
Δλ
ν
||
t
ν
|
k
a
n,ν1
n−1
ν1
a
n,ν1
|
Δλ
ν
|
k−1
.
4.9
||
4.10
holds. Thus by Lemma 3.3, 3.4 implies that
∞
n1
|Δλ
n
| converges. Therefore, there exists a
positive constant M such that
∞
n1
|Δλ
n
|≤M and from the properties of matrix A,weobtain
n−1
ν1
a
n,ν1
|
Δλ
k
|
≤ Ma
nn
. 4.11
We have, using v and x,
I
m
ν1
β
ν
|
t
ν
|
k
m1
nν1
n
δk
a
n,ν1
.
4.12
Therefore,
I
2
O
1
m
ν1
I
2
: O
1
m−1
ν1
Δ
νβ
ν
ν
r1
r
δk−1
|
t
r
|
k
O
1
mβ
m
1
m−1
ν1
β
ν1
X
ν1
O
1
O
1
.
4.14
Using H
¨
older’s inequality and viii,
m1
n2
n
k−1
|
T
k
≤
m1
n2
n
δkk−1
n−1
ν1
|
λ
ν1
|
a
n,ν1
ν
|
t
ν
|
k
O
1
m1
n−1
ν1
|
λ
ν1
|
k
a
νν
|
t
ν
|
k
a
n,ν1
n−1
ν1
a
νν
|
a
n,ν1
|
k−1
νν
|
t
ν
|
k
a
n,ν1
O
1
m
ν1
|
λ
ν1
|
a
νν
|
t
ν
|
k
m1
nν1
n
|
λ
v1
|
va
vv
v
δk−1
|
t
v
|
k
O
1
m
v1
|
λ
v1
|
v
δk−1
|
t
O
1
|
λ
m1
|
m
v1
v
δk−1
|
t
v
|
k
O
1
m−1
v1
|
Δλ
v1
|
v1
m−1
v1
|
Δλ
v1
|
X
v1
O
1
|
λ
m1
|
X
m1
O
1
m−1
v1
β
v1
X
m
n1
n
δkk−1
n 1
a
nn
λ
n
t
n
n
k
O
1
m
} 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
O
X
m
,m−→ ∞ , 5.1
are satisfied then the series
a
n
i0
p
i
→∞, as n →∞satisfies
np
n
O
P
n
, as n −→ ∞ , 5.2
m1
nv1
n
δk
p
n
P
n
P
n−1
O
v
δk
P
. Conditions i and ii of Corollary 5.3 are,
respectively, conditions i and ii of Theorem 2.1. Condition v becomes condition 5.2 and
conditions ix and x become condition 5.3 for weighted mean method. Conditions vi,
vii,andviii of Theorem 2.1 are automatically satisfied for any weighted mean method.
The following Corollary is the special case of Corollary 5.3 for μ 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
1 B. E. Rhoades and E. Savas¸, “A summability factor theorem for generalized absolute summability,”
Real Analysis Exchange, vol. 31, no. 2, pp. 355–363, 2006.
2 T. M. Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,”
Proceedings of the London Mathematical Society , vol. 7, pp. 113–141, 1957.
3 S. Alijancic and D. Arendelovic, “O-regularly varying functions,” Publications de l’Institut Math
´
ematique
, vol. 22, no. 36, pp. 5–22, 1977.
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