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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 =

n
∑
v=0

The series

∑

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, . . . ,
nann = O(1),

(1.8)
(1.9)

as n → ∞,

(1.10)

λn = o(m), m → ∞,

(1.11)

|∆λn | = o(m), m → ∞.

(1.12)

m

∑

n=1
m
∑

n=1

If (Xn ) is a quasi-f −increasing sequence satisfying
m
∑

k

n−1 |tn | = O(Xm ), m → ∞,

(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
∑

n=1

1

γ

n (nβ log nXn )

k−1

∞
∑

)
(
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
|tn | = O (Xm ) ,
|t
|
=
O
(1)
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
log nXn )

m−1
∑

( n
∑

1

+

v=1

m
∑

n=1

O(1)

µ

v (v β log vXv )

µ

n (nβ log nXn )

m−1
∑

k

k−1

k−1

(

nβ logµ nXn

|tv |

|tn |

k

(

k

)

)k−1

(
)k−1
∆ nβ logµ nXn

mβ logµ mXm

)k−1

 (
)k−1 

nβ logγ nXn ∆ nβ logµ nXn


+O mβ logγ mXm
(

|tn |

1

n=1

(

=

k−1

n=1

n=1

=

µ

n (nβ

β

)(

mβ logµ mXm

γ

O (m − 1) log (m − 1) Xm−1

)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 ||

O(1)

v
∑

∞
∑

v β+1 logµ vXv |∆ |∆λv ||

v=1

=

nβ logµ nXn

n=1

v=1

=

∆ |∆λv |

v=n

n=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
and (1.17),
then the series
an λn is summable |A|k , k ≥ 1, where (fn ) =
((1.15)
µ)
nβ (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

= Tn1 + Tn2 + Tn3 + Tn4 .

(3.2)

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

k−1

|Tn1 |

k

=

n=1

m
∑

n

k−1

n=1

=

O(1)

m
∑


k
n + 1



 n ann λn tn 

(nann )

n=1

=

O(1)

=

O(1)

O(1)

m
∑

m
∑

n=1

=

O(1)

m−1
∑

1
µ

n (nβ log nXn )

m−1
∑
n=1

=

O(1).

µ

m
∑

)k−1
(
k
|tn | |λn | |λn | nβ logµ nXn

k

log nXn )

|∆λn |

+O(1) |λm |

O(1)

k−1

1
n (nβ

n=1

=

1
k
k
|tn | |λn |
n

m
∑
1
k
k
|tn | |λn |
n
n=1

n=1

=

k

k−1

n
∑

|tn | |λn |

1

µ
β
v=1 v (v log vXv )

k−1

1

µ
β
n=1 n (n log nXn )

k−1

|tv |

|tn |

k

k

|∆λn | nβ logµ nXn + O(1) |λm | mβ logµ mXm

ON QUASI-POWER INCREASING SEQUENCES

67

As, (see[2]),

n−1
∑

|∆v ânv |

=

n−1
∑

(an−1,v − an,v ) = 1 − 1 + ann = ann ,

v=0

v=0

therefore, in view of (2.5),

m+1
∑

n

k−1

|Tn2 |

k

=

m+1
∑

n

k−1

n=2

n=2

≤

m+1
∑

v=1

n

k−1

O(1)

n−1
∑

k

|∆v ânv | |λv | |tv |

m+1
∑

(nann )

k−1

n=2

=

O(1)

m
∑

=

O(1)

k

|λv | |tv |

n−1
∑

k

m
∑

k

avv |λv | |tv |

k

|∆v ânv |

k

v=1

k

|λv | |tv |

k

=

O(1)

=

O(1), as in the case of Tn1 .

v
v=1

|∆v ânv |

|∆v ânv | |λv | |tv |

n=v+1

m
∑
1

(n−1
∑

v=1

v=1

m
∑

k

v=1

v=1

n=2

=

n−1
k
∑



∆v ânv λv tv 




k

)k−1

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
∑

v=1

n

k−1

O(1)

n−1
∑

m+1
∑

(ân,v+1 ) |∆λv |

n

k−1

O(1)

m
∑

|∆λv |

v=1

=

O(1)

m
∑

|∆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 |

O(1)

|tv |

r=1

v=1

m
∑
v=1

=

O(1)

m
∑

v
∑

β

k−1

k−1

=

O(1).

m+1
∑

ân,v+1 (nann )

k−1

ân,v+1

n=v+1

k−1

1
µ

r (rβ log rXr )
µ

log vXv )

k−1

m
∑
v=1

v=1

+O(1) |∆λm | m

k−1

k

µ

µ

k−1

n=v+1

|∆λv | v log vXv + O(1)
β+1

k−1

nk−1 ân,v+1 (ann )

m+1
∑

1
v (v β

k

nk−1 ân,v+1 (ân,v+1 )

m+1
∑

log mXm

k−1

|tr |

|tr |

µ

|∆λv | v log vXv

v=1

(v β logµ vXv )

m+1
∑

β

n=v+1

v (v β logµ vXv )

∆ (v |∆λv |)

+O(1)m |∆λm |

k−1

k

(v β logµ vXv )

(n−1
∑

n=v+1

k

(v β logµ vXv )

v |∆λv |

m−1
∑

log vXv )

|tv |

k−1

k

µ

|tv |

v=1

=

k

k−1

|tv |

k

(ân,v+1 ) |∆λv |

|tv |

(v β

k

(v β logµ vXv )

v=1

n=2

=

|tv |

k

v=1

n=2

=

n−1
k
∑



ân,v+1 ∆λv tv 




k

k

|∆ |∆λv || v β+1 logµ vXv

)k−1

ON QUASI-POWER INCREASING SEQUENCES
m+1
∑

nk−1 |Tn4 |

k

=

n=2

m+1
∑
n=2

=

m+1
∑

k
n−1

∑


nk−1 
v −1 ân,v λv tv 


v=1

n

k−1

O(1)

n−1
∑

(vavv )

O(1)

m+1
∑

(nann )

k−1

O(1)

avv ân,v |λv | |tv |

m
∑

n−1
∑

m
∑

k

(n−1
∑

avv ân,v

(vavv )

−1

k

avv ân,v |λv | |tv |

)k−1

k

v=1

k

avv |λv | |tv |

k

m+1
∑

ân,v

n=v+1

v=1

=

k

v=1

n=2

=

−k

v=1

n=2

=

69

k

avv |λv | |tv |

k

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