Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 2(2011), Pages 61-69.
ON QUASI-POWER INCREASING SEQUENCES GENERAL
CONTRACTIVE CONDITION OF INTEGRAL TYPE
(COMMUNICATED BY MOHAMMAD SAL MOSLEHIAN )
W.T. SULAIMAN
Abstract. A general result concerning absolute summability of infinite series
by quasi-power increasing sequence is proved. Our result gives three improvements to the result of Sevli and Leindler [4].
1. Introduction
∑
Let
an be an infinite
series with partial sım (sn ), A denote a lower triangular
∑
matrix. The series
an is said to be absolutely A-summable of order k ≥ 1, if
∞
∑
k
(1.1)
anv sv .
(1.2)
nk−1 |Tn − Tn−1 | < ∞,
n=1
where
Tn =
The series
∑
n
∑
v=0
an is summable |A|k , k ≥ 1, if
∞
∑
k
nk−1 |Tn − Tn−1 | < ∞.
(1.3)
n=1
Let tn denote the nth (C, 1) mean of the sequence (nan ), i,e.,
tn =
n
1 ∑
vav .
n + 1 v=1
A positive sequence γ = (γn ) is said to be a quasi−β−power increasing sequence if
there exists a constant K = K(β, γ) ≥ 1 such that
Knβ γn ≥ mβ γm
2000 Mathematics Subject Classification. 40F35, 40D25.
Key words and phrases. Absolute summability, Quasi-power increasing sequence.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted January 15, 2011. Published May 4, 2011.
61
(1.4)
62
W.T. SULAIMAN
holds for all n ≥ m ≥ 1. It may be mentioned that every almost increasing sequence
is a quasi-β−power increasing sequence for any nonnegative β, but the converse need
not be true.
A positive sequence γ = (γn ) is said to be a quasi−f −power increasing sequence
if (see[5]) there exists a constant K = K(γ, f ) ≥ 1 such that
Kfn γn ≥ fm γm
(1.5)
holds for all n ≥ m ≥ 1.
Two lower triangular matrices A and  are associated with A as follows
anv =
n
∑
anr ,
n, v = 0, 1, . . . ,
(1.6)
r=v
ânv = anv − an−1,v , n = 1, 2, . . . ,
â00 = a00 = a00 .
(1.7)
Sevli and Leindler [4] proved the following result
Theorem 1.1. Let A be lower triangular matrix with nonnegative entries satisfying
an−1,v ≥ an,v
for n ≥ v + 1,
an0 = 1, n = 0, 1, . . . ,
(1.8)
(1.9)
as n → ∞,
(1.10)
λn = o(m), m → ∞,
(1.11)
|∆λn | = o(m), m → ∞.
(1.12)
nann = O(1),
m
∑
n=1
m
∑
n=1
If (Xn ) is a quasi-f −increasing sequence satisfying
m
∑
n−1 |tn | = O(Xm ), m → ∞,
k
(1.13)
n=1
∞
∑
nXn (β, µ) |∆ |∆λn || < ∞,
(1.14)
n=1
( β
∑
µ)
then the series an λn is summable
( β|A|k , k µ≥ 1, where)(fn ) = n (log n) , µ ≥ 0,
0 ≤ β < 1, and Xn (β, µ) = max n (log n) Xn , log n .We name the conditions
m
∑
1
k−1
γ
n=1
n (nβ log nXn )
∞
∑
(
)
k
|tn | = O mβ logγ mXm , m → ∞,
(1.15)
λn → 0, as n → ∞,
(1.16)
nβ+1 logγ nXn |∆ |∆λn || < ∞,
(1.17)
n=1
n−1
∑
v=1
avv ân,v = O(ann ),
1/nann = O(1).
(1.18)
ON QUASI-POWER INCREASING SEQUENCES
63
2. Lemmas
Lemma 2.1. [1]. Let A be as defined in theorem 1.1, then
ân,v+1 ≤ ann for n ≥ v + 1,
and
m+1
∑
ân,v+1 ≤ 1,
(2.1)
v = 0, 1, . . . .
(2.2)
n=v+1
Lemma 2.2. Condition (1.15) is weaker than (1.13).
Proof. Suppose that (1.13) is satisfied. Since nβ logγ nXn is non-decreasing, then
m
m
∑
∑
1
1
k
k
|t
|
=
O
(1)
|tn | = O (Xm ) ,
n
k−1
γ
β log nX )
n
n
(n
n
n=1
n=1
while if (1.15) is satisfied, we have
m
∑
1
k
|tn |
n
n=1
=
=
m
∑
1
µ
k−1
n=1
n (nβ
log nXn )
m−1
∑
( n
∑
1
+
v=1
k−1
v (v β log vXv )
m
∑
1
µ
n=1
= O(1)
n (nβ log nXn )
m−1
∑
k
(
nβ logµ nXn
)
µ
n=1
|tn |
k−1
|tv |
|tn |
k
(
k
)k−1
(
)k−1
∆ nβ logµ nXn
mβ logµ mXm
)k−1
(
)k−1 nβ logγ nXn ∆ nβ logµ nXn
n=1
(
+O mβ logγ mXm
(
β
)(
mβ logµ mXm
= O (m − 1) log (m − 1) Xm−1
γ
)k−1
) m−1
∑ ((
β
(n + 1) logµ (n + 1)Xn+1
n=1
(
(
)k−1 )
)k
− nβ logµ nXn
+ O mβ logµ mXm
(
)(
)k−1
(
)k
= O mβ logµ mXm mβ logµ mXm
+ O mβ logµ mXm
(
)k
= O mβ logµ mXm ̸= O (Xm ) .
Therefore (1.15) implies (1.13) but not conversely.
)k−1
64
W.T. SULAIMAN
Lemma 2.3. Condition (1.16) and (1.17) imply
mβ+1 logµ mXm |∆λm | = O(1),
∞
∑
m→∞
nβ logµ nXn |∆λn | = O(1),
(2.3)
(2.4)
n=1
and
nβ logµ nXn |λn | = O(1),
n → ∞.
(2.5)
Proof. As ∆λn → 0, and nβ logµ nXn is non-decreasing, we have
nβ+1 logµ nXn |∆λn | = nβ+1 logµ nXn
∞
∑
∆ |∆λv |
v=n
= O(1)
∞
∑
v β+1 logµ vXv |∆ |∆λv ||
v=n
= O(1).
This proves (2.3). To prove (2.4), we observe that
∞
∑
nβ logµ nXn |∆λn | =
n=1
∞
∑
nβ logµ nXn
∞
∑
∆ |∆λv |
v=n
n=1
≤
∞
∑
|∆ |∆λv ||
v=1
= O(1)
v
∑
nβ logµ nXn
n=1
∞
∑
v β+1 logµ vXv |∆ |∆λv ||
v=1
= O(1), by (1.17).
ON QUASI-POWER INCREASING SEQUENCES
65
Finally
nβ logµ nXn |λn | = nβ logµ nXn
∞
∑
∆ |λv |
v=n
≤
∞
∑
v β logµ vXv |∆λv |
v=n
= O(1), by (2.4).
3. Main Result
Theorem 3.1. Let A satisfies conditions (1.8)-(1.10) and (1.18), let (λn ) be a
sequence satisfies (1.16). If (Xn ) is∑a quasi-f -power increasing sequence satisfying
(1.15)
and (1.17),
then the series
an λn is summable |A|k , k ≥ 1, where (fn ) =
( β
µ)
n (log n) , µ ≥ 0, 0 ≤ β < 1.
Proof. Let xn be the nth term of the A-transform of the series
tion, we have
xn =
n
∑
anv sv =
v=0
n
∑
∑
an λn . By defini-
anv λv av ,
v=0
and hence
Tn := xn − xn−1 =
n
∑
v −1 ânv λv vav .
(3.1)
v=0
Applying Abel’s transformation,
Tn =
n−1
n−1
n−1
∑
∑
∑
n+1
ann λn tn +
∆v ânv λv tv +
ân,v+1 ∆λv tv +
v −1 ân,v λv tv
n
v=1
v=1
v=1
(3.2)
= Tn1 + Tn2 + Tn3 + Tn4 .
To complete the proof, by Minkowski’s inequality, it is sufficient to show that
∞
∑
n=1
nk−1 |Tnj |
k
< ∞,
j = 1, 2, 3, 4.
66
W.T. SULAIMAN
Applying Holder’s inequality, we have, in view of (2.5),
m
∑
n=1
n
k−1
|Tn1 |
k
=
m
∑
n
k−1
n=1
= O(1)
m
∑
k
n + 1
n ann λn tn k
(nann )
n=1
= O(1)
= O(1)
m
∑
1
k
k
|tn | |λn |
n
n=1
m
∑
= O(1)
n (nβ log nXn )
m
∑
n=1
m−1
∑
n=1
= O(1).
µ
k−1
|tn | |λn |
k
log nXn )
|∆λn |
+O(1) |λm |
m−1
∑
(
)k−1
k
|tn | |λn | |λn | nβ logµ nXn
1
n (nβ
n=1
= O(1)
k−1
1
µ
n=1
= O(1)
1
k
k
|tn | |λn |
n
m
∑
n
∑
1
µ
β
v=1 v (v log vXv )
k−1
1
|tv |
k
k
k−1
µ
β
n=1 n (n log nXn )
|tn |
|∆λn | nβ logµ nXn + O(1) |λm | mβ logµ mXm
ON QUASI-POWER INCREASING SEQUENCES
67
As, (see[2]),
n−1
∑
|∆v ânv | =
v=0
n−1
∑
(an−1,v − an,v ) = 1 − 1 + ann = ann ,
v=0
therefore, in view of (2.5),
m+1
∑
n
k−1
|Tn2 |
k
=
n=2
m+1
∑
n
k−1
n=2
≤
m+1
∑
n−1
k
∑
∆v ânv λv tv v=1
n
k−1
n=2
= O(1)
n−1
∑
k
|∆v ânv | |λv | |tv |
m+1
∑
m
∑
m
∑
(nann )
k−1
n−1
∑
k
v=1
k
|λv | |tv |
m
∑
k
|∆v ânv |
n=v+1
k
k
avv |λv | |tv |
m
∑
1
v
v=1
|∆v ânv |
|∆v ânv | |λv | |tv |
v=1
= O(1)
)k−1
v=1
v=1
= O(1)
(n−1
∑
v=1
n=2
= O(1)
k
k
|λv | |tv |
k
= O(1), as in the case of Tn1 .
k
68
W.T. SULAIMAN
In view of (2.4), (1.10), and (2.2),
m+1
∑
n
k−1
|Tn3 |
k
=
n=2
m+1
∑
n
k−1
n=2
≤
m+1
∑
n−1
k
∑
ân,v+1 ∆λv tv v=1
n
k−1
m+1
∑
n
(ân,v+1 ) |∆λv |
k−1
n=2
= O(1)
m
∑
= O(1)
|∆λv |
|∆λv |
v=1
= O(1)
m
∑
|∆λv |
v=1
= O(1)
m
∑
|∆λv |
v=1
= O(1)
m
∑
n−1
∑
(v β logµ vXv )
|tv |
(v β
|tv |
|tv |
m−1
∑
v=1
= O(1)
n=v+1
m+1
∑
n=v+1
k
1
r (rβ log rXr )
1
v (v β
µ
µ
m
∑
v=1
β+1
µ
log mXm
|tr |
k
k−1
log vXv )
|∆λv | v log vXv + O(1)
β
k
k−1
µ
v=1
= O(1).
ân,v+1
k−1
v
∑
r=1
+O(1) |∆λm | m
k−1
k−1
m
∑
v=1
m
∑
ân,v+1 (nann )
k−1
v (v β logµ vXv )
∆ (v |∆λv |)
+O(1)m |∆λm |
k−1
n=v+1
k
|tv |
nk−1 ân,v+1 (ann )
k−1
m+1
∑
(v β logµ vXv )
v |∆λv |
k−1
n=v+1
k
(v β logµ vXv )
k
nk−1 ân,v+1 (ân,v+1 )
m+1
∑
log vXv )
|∆λv | v log vXv
v=1
k−1
k−1
k
µ
)k−1
β
(v β logµ vXv )
m+1
∑
k
k−1
|tv |
k
(ân,v+1 ) |∆λv |
|tv |
v=1
= O(1)
(v β logµ vXv )
v=1
v=1
m
∑
(n−1
∑
k
|tv |
k
v=1
n=2
= O(1)
n−1
∑
|tr |
|∆ |∆λv || v β+1 logµ vXv
µ
ON QUASI-POWER INCREASING SEQUENCES
m+1
∑
nk−1 |Tn4 |
k
=
n=2
m+1
∑
n=2
=
m+1
∑
69
n−1
k
∑
nk−1 v −1 ân,v λv tv v=1
n
k−1
n=2
= O(1)
n−1
∑
−k
(vavv )
m+1
∑
m
∑
m
∑
k
(n−1
∑
)k−1
avv ân,v
v=1
(nann )
k−1
n−1
∑
−1
(vavv )
k
k
avv ân,v |λv | |tv |
v=1
k
k
avv |λv | |tv |
v=1
= O(1)
avv ân,v |λv | |tv |
v=1
n=2
= O(1)
k
m+1
∑
ân,v
n=v+1
k
k
avv |λv | |tv |
v=1
= O(1), as in the case of Tn1 .
Remark 3.2. As an applications to theorem 3.1, improvements to all corollaries
mentioned in [4] can also obtained via weaker conditions
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
References
[1] B.E. Rhodes, Inclusion theorems for absolute matrix summability methods,J. Math. Anal.
Appl. 238 , (1999), 87-90.
[2] E. Savas, Quasi-power increasing sequence for generalized absolute summability, Nonlinear
Analysis, 68 (2008), 170-176.
[3] E. Savas and H. Sevli, A recent note on quasi-power increasing sequence for generalized absolute summability, J. Ineq. Appl., (2009)Art. Id:675403.
[4] H. Sevli and L. Leindler, On the absolute summability factors of infinite series involving
quasi-power increasing sequences, Computers and mathematics with applications 57 (2009)
702–709.
[5] W.T. Sulaiman, Extension on absolute summability factors of infinite series, J. Math. Anal.
Appl. 322 (2), (2006), 1224-1230.
Waad Sulaiman, Department of Computer Engineering, College of Engineering, University of Mosul, Iraq.
E-mail address: waadsulaiman@hotmail.com