Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 1 Issue 3(2009), Pages 1-9.
A STUDY ON ABSOLUTE FACTORS FOR A TRIANGULAR
MATRIX
(DEDICATED IN OCCASION OF THE 65-YEARS OF
PROFESSOR R.K. RAINA)
W. T. SULAIMAN
Abstract. In this paper a result concerning summability factors theorem for
lower triangular matrices is presented. This result generalized and extend the
result of Savas [1].
1. Introduction
Let 𝑇 = (𝑡𝑛𝑣 ) be a lower
∑ triangular matrix, (𝑠𝑛 ) be the sequence of the 𝑛-th
partial sums of the series
𝑎𝑛 , then, we define
𝑇𝑛 :=
𝑛
∑
𝑡𝑛𝑣 𝑠𝑣 .
(1.1)
𝑣=0
A series
∑
𝑎𝑛 is said to be summable ∣ 𝑇 ∣𝑘 , 𝑘 ≥ 1, if
∞
∑
𝑛𝑘−1 ∣ Δ𝑇𝑛−1 ∣𝑘 < ∞.
(1.2)
𝑛=1
One of the most interesting cases of ∣ 𝑇 ∣𝑘 -summability is the case when 𝑇 is chosen
to be the Cesaro matrix. That is by putting
𝑡𝑛𝑣 =
𝐴𝛼−1
Γ(𝑛 + 𝛼 + 1)
𝑛−𝑣
, 𝑣 = 0, 1, ..., 𝑛, 𝐴𝛼
, 𝑛 = 0, 1, ... .
𝑛 =
𝐴𝛼
Γ(𝛼
+ 1)Γ(𝑛 + 1)
𝑛
Given any lower triangular matrix 𝑇 one can associate the matrices 𝑇 and 𝑇ˆ, with
entries defined by
𝑡𝑛𝑣 =
𝑛
∑
𝑡𝑛𝑖 , 𝑛 𝑎𝑛𝑑 𝑖 = 0, 1, 2, ..., ˆ
𝑡𝑛𝑣 = 𝑡𝑛𝑣 − 𝑡𝑛−1,𝑣 ,
𝑖=𝑣
2000 Mathematics Subject Classification. 40F05, 40D25.
Key words and phrases. Absolute summability; summability factor; weight function.
c
⃝2009
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted Jun, 2009. Published September, 2009.
1
2
W. T. SULAIMAN
respectively. With 𝑠𝑛 =
𝑡𝑛 :=
𝑛
∑
𝑡𝑛𝑣 𝑠𝑣 =
𝑣=0
∑𝑛
𝑖=0
𝑛
∑
𝑎𝑖 𝜆𝑖 , we define and derive the following
𝑡𝑛𝑣
𝑣=0
𝑌𝑛 := 𝑡𝑛 − 𝑡𝑛−1 =
𝑛
∑
𝑣
∑
𝑖=0
𝑡𝑛𝑖 𝑎𝑖 𝜆𝑖 −
𝑖=0
𝑎𝑖 𝜆𝑖 =
𝑛−1
∑
𝑛
∑
𝑛
∑
𝑎𝑖 𝜆𝑖
𝑖=0
𝑡𝑛𝑣 =
𝑣=𝑖
𝑡𝑛−1,𝑖 𝑎𝑖 𝜆𝑖 =
𝑛
∑
𝑖=0
𝑛
∑
𝑡𝑛𝑖 𝑎𝑖 𝜆𝑖 ,
(1.3)
𝑖=0
ˆ
𝑡𝑛𝑖 𝑎𝑖 𝜆𝑖 , 𝑎𝑠 𝑡𝑛−1,𝑛 = 0. (1.4)
𝑖=0
𝑋𝑛 := 𝑢𝑛 − 𝑢𝑛−1 =
𝑛
∑
𝑢
ˆ𝑛𝑖 𝑎𝑖 𝜇𝑖 .
(1.5)
𝑖=0
We call 𝑇 a triangle if 𝑇 is lower triangular and 𝑡𝑛𝑛 ∕= 0 for all 𝑛. A triangle 𝐴 is
called factorable if its nonzero entries 𝑎𝑚𝑛 can be written in the form 𝑏𝑚 𝑐𝑛 for each
𝑚 and 𝑛. Recall that ˆ
𝑡𝑛𝑛 = 𝑡𝑛𝑛 . We also assume that (𝑝𝑛 ) is a positive sequence
of numbers such that
𝑃𝑛 = 𝑝0 + 𝑝1 + ... + 𝑝𝑛 → ∞, 𝑎𝑠 𝑛 → ∞.
A positive sequence (𝑎𝑛 ) is said to be almost increasing if 𝐴𝑏𝑛 ≤ 𝑎𝑛 ≤ 𝐵𝑏𝑛 ,
where (𝑏𝑛 ) is a positive increasing sequence and 𝐴 and 𝐵 are positive constants
(see [1]).
A positive sequence (𝛾𝑛 ) is said to be quasi 𝛽−power increasing sequence if there
is a constant 𝐾 = 𝐾(𝛽, 𝛾) ≥ 1 such that 𝐾𝑛𝛽 𝛾𝑛 ≥ 𝑚𝛽 𝛾𝑚 holds for all 𝑛 ≥ 𝑚 ≥ 1.
It should be mentioned that every almost increasing sequence is quasi 𝛽−power increasing sequence for any 𝛽 > 0, while the converse need not be true as for example
𝛾𝑛 = 𝑛−𝛽 , 𝛽 > 0.
Here we generalize the quasi 𝛽-power increasing sequence by giving the following.
Definition.
A sequence (𝜙𝑛 ) is said to be quasi (𝛽 − 𝛾)−power increasing sequence if there
exist 𝐾 = 𝐾(𝛽, 𝛾) ≥ 1 such that 𝐾𝑛𝛽 (log 𝑛)𝛾 𝜙𝑛 ≥ 𝑚𝛽 (log 𝑚)𝛾 𝜙𝑚 holds for all
𝑛 ≥ 𝑚 ≥ 1, 𝛾 ≥ 0. Clearly quasi (𝛽 − 0)−power increasing sequence is the quasi 𝛽−
power increasing sequence.
The series
∑
𝑎𝑛 is said to be summable ∣ 𝑅, 𝑝𝑛 ∣𝑘 , 𝑘 ≥ 1, if
∞
∑
𝑛𝑘−1 ∣ 𝜎𝑛 − 𝜎𝑛−1 ∣𝑘 < ∞,
𝑛=1
where
𝜎𝑛 =
𝑛
1 ∑
𝑝𝑣 𝑠𝑣 .
𝑃𝑛 𝑣=0
The (𝐶, 1)−mean of the sequence (𝑓𝑛 ) is equal to
𝑓 (𝑛) = 𝑂(𝑔(𝑛)) if
(𝑛)
lim𝑛→∞ 𝑓𝑔(𝑛)
1
𝑛+1
∑𝑛
= 𝑘 < ∞.
Very recently, Savas [1] proved the following theorem.
𝑣=0
𝑓𝑣 , and we said that
A STUDY ON ABSOLUTE FACTORS FOR A TRIANGULAR MATRIX
3
Theorem 1.1. Let 𝐴 be a lower triangular matrix with nonegative entries such
that
(i) 𝑎𝑛0 = 1, 𝑛 = 0, 1, ...,
(ii) 𝑎𝑛−1,𝑣 ≥ 𝑎𝑛𝑣 𝑓 𝑜𝑟 𝑛 ≥ 𝑣 + 1,
(iii)𝑛𝑎𝑛𝑛 = 𝑂(1), 1 = 𝑂(𝑛𝑎𝑛𝑛 ),
∑𝑛−1
(iv) 𝑣=1 𝑎𝑣𝑣 ∣ ˆ
𝑎𝑛,𝑣+1 ∣= 𝑂(𝑎𝑛𝑛 ).
Let ∑
𝑡1𝑛 denote the 𝑛−th (𝐶, 1) mean of (𝑛𝑎𝑛 ). If
∞
(v) ∑𝑣=1 𝑎𝑣𝑣 ∣ 𝜆𝑣 ∣𝑘 ∣ 𝑡1𝑣 ∣𝑘 = 𝑂(1),
∞
∣1−𝑘 ∣ Δ𝜆𝑣 ∣𝑘 ∣ 𝑡1𝑣 ∣𝑘 = 𝑂(1),
(vi) 𝑣=1 ∣ 𝑎𝑣𝑣 ∑
then the series
𝑎𝑛 𝜆𝑛 is ∣ 𝐴 ∣𝑘 summable, 𝑘 ∈ 𝑁.
The aim of this paper is to give the following three improvements to the previous
result (see Theorem 2.):
1. The lower triangular matrix we assumed is not restricted to nonnegative entries.
2. The kind of summability we obtained is ∣ 𝐴, 𝛿 ∣𝑘 , 𝛿 ≥ 0, in which ∣ 𝐴 ∣𝑘 ≡∣ 𝐴, 0 ∣𝑘 ,
and 𝑘 is not restricted to integers but 𝑘 ∈ [1, ∞).
3. An extension is made by assuming further hypothesis.
2. Results.
The following is our main result.
Theorem 2.1. Let 𝑇 be a lower triangular matrix, 𝑡𝑛 denote the 𝑛−th (𝐶, 1) mean
of the sequence (𝑛𝑎𝑛 ), and let (𝜆𝑛 ) be a sequence of numbers such that 𝑇, 𝑡𝑛 , 𝜆𝑛 are
all satisfying
𝑛 ∣ 𝑡𝑛𝑛 ∣= 𝑂(1), 1 = 𝑂(𝑛 ∣ 𝑡𝑛𝑛 ∣),
(2.1)
𝑛−1
∑
∣ 𝑡𝑣𝑣 ∣∣ ˆ
𝑡𝑛𝑣 ∣= 𝑂(∣ 𝑡𝑛𝑛 ∣),
(2.2)
∣ 𝑡𝑣𝑣 ∣∣ ˆ
𝑡𝑛,𝑣+1 ∣= 𝑂(∣ 𝑡𝑛𝑛 ∣),
(2.3)
𝑣=1
𝑛−1
∑
𝑣=1
∞
∑
𝑛𝛿𝑘 ∣ ˆ
𝑡𝑛𝑣 ∣= 𝑂(𝑣 𝛿𝑘 ),
(2.4)
∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣= 𝑂(∣ 𝑡𝑛𝑛 ∣),
(2.5)
𝑛=𝑣+1
𝑛−1
∑
𝑣=1
∞
∑
𝑛𝛿𝑘 ∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣= 𝑂(𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣),
(2.6)
𝑛=𝑣+1
∞
∑
𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1),
(2.7)
𝑣 𝛿𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ Δ𝜆𝑣 ∣𝑘 = 𝑂(1),
(2.8)
𝑣=1
∞
∑
𝑣=1
4
W. T. SULAIMAN
∑
then the series
𝑎𝑛 𝜆𝑛 is summable ∣ 𝑇, 𝛿 ∣𝑘 , 𝑘 ≥ 1, 𝛿 ≥ 0.
Furthermore if (𝑋𝑛 ) is a quasi (𝛽−𝛾)−power increasing sequence, 0 < 𝛽 < 1, 𝛾 ≥ 0,
and if (𝛽𝑛 ) is a sequence of numbers satisfying
Δ𝜆𝑛 ≤ 𝛽,
(2.9)
𝛽𝑛 → 0 𝑎𝑠 𝑛 → ∞,
(2.10)
∞
∑
𝑛 ∣ Δ𝛽𝑛 ∣ 𝑋𝑛 < ∞,
(2.11)
𝑛=1
∣ 𝜆𝑛 ∣ 𝑋𝑛 = 𝑂(1),
(2.12)
𝑚
∑
𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘
= 𝑂(𝑋𝑚 ),
𝑋𝑣𝑘−1
𝑣=1
(2.13)
then the conditions
∑ (2.7) and (2.8) are omitted in order to obtain the ∣ 𝑇, 𝛿 ∣𝑘
−summability of
𝑎𝑛 𝜆𝑛 .
Proof of Theorem 2.1. We have
𝑛
∑
𝑛
∑
ˆ
𝑡𝑛𝑣 𝜆𝑣
𝑣
𝑣=0
𝑣=1
(
)
(
) ( 𝑛
)(
)
𝑛−1
𝑣
∑ ∑
∑
ˆ
𝑡𝑛𝑣 𝜆𝑣
𝑡𝑛𝑛 𝜆𝑛
𝑟𝑎𝑟 Δ𝑣
=
+
𝑣𝑎𝑣
𝑣
𝑛
𝑣=1 𝑟=1
𝑣=1
(
)
𝑛−1
∑
1
1
1 ˆ
𝑛+1
ˆ
=
(𝑣 + 1)𝑡𝑣
𝑡𝑛𝑣 𝜆𝑣 +
Δˆ
𝑡𝑛𝑣 𝜆𝑣 +
𝑡𝑛,𝑣+1 Δ𝜆𝑣 +
𝑡𝑛 𝑡𝑛𝑛 𝜆𝑛
𝑣(𝑣 + 1)
𝑣+1
𝑣+1
𝑛
𝑣=1
)
𝑛−1
∑ (1
𝑛+1
ˆ
ˆ
ˆ
=
𝑡𝑣 𝑡𝑛𝑣 𝜆𝑣 + 𝑡𝑣 Δ𝑣 𝑡𝑛𝑣 𝜆𝑣 + 𝑡𝑣 𝑡𝑛,𝑣+1 Δ𝜆𝑣 +
𝑡𝑛 𝑡𝑛𝑛 𝜆𝑛
𝑣
𝑛
𝑣=1
𝑌𝑛 :=
ˆ
𝑡𝑛𝑣 𝜆𝑣 𝑎𝑣 =
𝑣𝑎𝑣
= 𝑌𝑛1 + 𝑌𝑛2 + 𝑌𝑛3 + 𝑌𝑛4 .
In order to prove the Theorem, by Minkowski’s inequality, we have to show that
∞
∑
𝑛=1
𝑛𝛿𝑘+𝑘−1 ∣ 𝑌𝑛𝑗 ∣𝑘 < ∞,
𝑗 = 1, 2, 3, 4.
A STUDY ON ABSOLUTE FACTORS FOR A TRIANGULAR MATRIX
5
By Holder’s inequality
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣ 𝑌𝑛1 ∣𝑘 =
𝑛=2
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣
𝑛=2
≤
∞
∑
𝑛−1
∑
𝑣=1
𝑛
𝛿𝑘+𝑘−1
𝑛=2
= 𝑂(1)
𝑛−1
∑
𝑣
1 ˆ
𝑡𝑣 𝑡𝑛𝑣 𝜆𝑣 ∣𝑘
𝑣
−𝑘
∣ 𝑡𝑣𝑣 ∣
𝑘
∣ 𝑡𝑣 ∣ ∣ ˆ
𝑡𝑛𝑣 ∣∣ 𝜆𝑣 ∣
𝑣=1
∞
∑
∞
∑
𝑛𝛿𝑘 (𝑛 ∣ 𝑡𝑛𝑛 ∣)𝑘−1
𝑛−1
∑
= 𝑂(1)
)𝑘−1
(𝑛−1
∑
∣ 𝑡𝑣𝑣 ∣∣ ˆ
𝑡𝑛𝑣 ∣
𝑣 −𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ ˆ
𝑡𝑛𝑣 ∣∣ 𝜆𝑣 ∣𝑘
𝑣=1
∞
∑
𝑣 −𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘
𝑣=1
∞
∑
𝑘
𝑣=1
𝑛=2
= 𝑂(1)
1−𝑘
𝑛𝛿𝑘 ∣ ˆ
𝑡𝑛𝑣 ∣
𝑛=𝑣+1
𝑣 𝛿𝑘−𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘
𝑣=1
= 𝑂(1)
∞
∑
𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1),
𝑣=1
in view of (2.2) and (2.7).
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣ 𝑌𝑛2 ∣𝑘 =
𝑛=2
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣
𝑛=2
≤
∞
∑
𝑛−1
∑
𝑡𝑣 Δ𝑣 ˆ
𝑡𝑛𝑣 𝜆𝑣 ∣𝑘
𝑣=1
𝑛
𝛿𝑘+𝑘−1
𝑛=2
= 𝑂(1)
𝑛−1
∑
∞
∑
∞
∑
𝑣=1
= 𝑂(1)
𝑘
∣ 𝑡𝑣 ∣ ∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣∣ 𝜆𝑣 ∣
(𝑛−1
∑
𝑣=1
∞
∑
𝑣=1
𝑛−1
∑
∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣
∣ 𝑡𝑣 ∣𝑘 ∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣∣ 𝜆𝑣 ∣𝑘
𝑣=1
∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘
)𝑘−1
𝑣=1
𝑛𝛿𝑘 (𝑛 ∣ 𝑡𝑛𝑛 ∣)𝑘−1
𝑛=2
= 𝑂(1)
𝑘
∞
∑
𝑛𝛿𝑘 ∣ Δ𝑣 ˆ
𝑡𝑛𝑣 ∣
𝑛=𝑣+1
𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1),
6
W. T. SULAIMAN
in view of (2.5), (2.6) and (2.7).
∞
∑
𝑛
𝛿𝑘+𝑘−1
𝑘
∣ 𝑌𝑛3 ∣ =
𝑛=2
∞
∑
𝑛
𝛿𝑘+𝑘−1
∣
𝑛−1
∑
𝑛=2
≤
∞
∑
𝑡𝑣 ˆ
𝑡𝑛,𝑣+1 Δ𝜆𝑣 ∣𝑘
𝑣=1
𝑛
𝛿𝑘+𝑘−1
𝑛=2
= 𝑂(1)
𝑛−1
∑
𝑘
∣ 𝑡𝑣 ∣ ∣ 𝑡𝑣𝑣 ∣
(𝑛−1
∑
∞
∑
∞
∑
∞
∑
)𝑘−1
∣ 𝑡𝑣𝑣
∣∣ ˆ
𝑡𝑛,𝑣+1 ∣
𝑣=1
𝑛𝛿𝑘 (𝑛 ∣ 𝑡𝑛𝑛 ∣)𝑘−1
𝑛−1
∑
∣ 𝑡𝑣 ∣𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ ˆ
𝑡𝑛,𝑣+1 ∣∣ Δ𝜆𝑣 ∣𝑘
𝑣=1
∣ 𝑡𝑣 ∣𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ Δ𝜆𝑣 ∣𝑘
𝑣=1
= 𝑂(1)
∣ˆ
𝑡𝑛,𝑣+1 ∣∣ Δ𝜆𝑣 ∣𝑘
𝑣=1
𝑛=2
= 𝑂(1)
1−𝑘
∞
∑
𝑛𝛿𝑘 ∣ ˆ
𝑡𝑛,𝑣+1 ∣
𝑛=𝑣+1
𝑣 𝛿𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ Δ𝜆𝑣 ∣𝑘 = 𝑂(1),
𝑣=1
in view of (2.3), (2.4) and (2.8).
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣ 𝑌𝑛4 ∣𝑘 =
𝑛=2
∞
∑
𝑛𝛿𝑘+𝑘−1 ∣
𝑛=2
= 𝑂(1)
= 𝑂(1)
∞
∑
𝑛=1
∞
∑
𝑛+1
𝑡𝑛 𝑡𝑛𝑛 𝜆𝑛 ∣𝑘
𝑛
𝑛𝛿𝑘+𝑘−1 ∣ 𝑡𝑛 ∣𝑘 ∣ 𝑡𝑛𝑛 ∣𝑘 ∣ 𝜆𝑛 ∣𝑘
𝑛𝛿𝑘 ∣ 𝑡𝑛𝑛 ∣∣ 𝑡𝑛 ∣𝑘 ∣ 𝜆𝑛 ∣𝑘 = 𝑂(1),
𝑛=1
in view of (2.7).
In order to complete the proof the rest, it is sufficient to show that
∞
∑
𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1),
𝑣=1
∞
∑
𝑣 𝛿𝑘 ∣ 𝑡𝑣 ∣𝑘 ∣ 𝑡𝑣𝑣 ∣1−𝑘 ∣ Δ𝜆𝑣 ∣𝑘 = 𝑂(1).
𝑣=1
Now,
𝑚
𝑚
∑
∑
𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘 𝑘−1
𝑣 𝛿𝑘 ∣ 𝑡𝑣𝑣 ∣∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1)
𝑋𝑣 ∣ 𝜆𝑣 ∣𝑘−1 ∣ 𝜆𝑣 ∣
𝑘−1
𝑋
𝑣
𝑣=1
𝑣=1
𝑚
∑
𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘
∣ 𝜆𝑣 ∣
𝑋𝑣𝑘−1
𝑣=1
( 𝑣
)
(𝑚
)
𝑚−1
∑ 𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘
∑ ∑
𝑟𝛿𝑘−1 ∣ 𝑡𝑟 ∣𝑘
∣ Δ ∣ 𝜆𝑣 ∣ ∣ +𝑂(1)
∣ 𝜆𝑚 ∣
= 𝑂(1)
𝑋𝑟𝑘−1
𝑋𝑣𝑘−1
𝑣=1
𝑟=1
𝑣=1
= 𝑂(1)
= 𝑂(1)
𝑚−1
∑
𝑋𝑣 ∣ Δ𝜆𝑣 ∣ +𝑂(1)𝑋𝑚 ∣ 𝜆𝑚 ∣
𝑣=1
= 𝑂(1)
𝑚−1
∑
𝑣=1
𝑋𝑣 𝛽𝑣 + 𝑂(1)𝑋𝑚 ∣ 𝜆𝑚 ∣= 𝑂(1).
A STUDY ON ABSOLUTE FACTORS FOR A TRIANGULAR MATRIX
𝑚
∑
𝑣
𝛿𝑘
𝑘
1−𝑘
∣ 𝑡𝑣 ∣ ∣ 𝑡𝑣𝑣 ∣
𝑘
∣ Δ𝜆𝑣 ∣ = 𝑂(1)
𝑣=1
𝑚
∑
7
𝑣 𝛿𝑘+𝑘−1 ∣ 𝑡𝑣 ∣𝑘 ∣ Δ𝜆𝑣 ∣𝑘
𝑣=1
𝑚
∑
𝑣 𝛿𝑘 ∣ 𝑡𝑣 ∣𝑘
= 𝑂(1)
(𝑣𝑋𝑣 𝛽𝑣 )𝑘−1 𝛽𝑣
𝑘−1
𝑋
𝑣
𝑣=1
𝑚
∑
𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘
𝑣𝛽𝑣
𝑋𝑣𝑘−1
𝑣=1
)
(𝑚
)
( 𝑣
𝑚−1
∑ 𝑣 𝛿𝑘−1 ∣ 𝑡𝑣 ∣𝑘
∑ ∑
𝑟𝛿𝑘−1 ∣ 𝑡𝑟 ∣𝑘
∣ Δ(𝑣𝛽𝑣 ) ∣ +𝑂(1)
𝑚𝛽𝑚
= 𝑂(1)
𝑋𝑟𝑘−1
𝑋𝑣𝑘−1
𝑣=1
𝑣=1
𝑟=1
= 𝑂(1)
= 𝑂(1)
𝑚
∑
𝑋𝑣 𝛽𝑣 + 𝑂(1)
𝑣=1
𝑚
∑
𝑣𝑋𝑣 ∣ Δ𝛽𝑣 ∣ +𝑂(1)𝑚𝑋𝑚 𝛽𝑚
𝑣=1
= 𝑂(1).
This completes the proof of the Theorem.
Recalling Lemma 2.2 and its proof.
Lemma 2.2. Let (𝑋𝑛 ) be a quasi (𝛽 − 𝛾)−power increasing sequence, 0 < 𝛽 <
1, 𝛾 ≥ 0, then the conditions (2.10) and (2.11) implies
∞
∑
𝛽𝑛 𝑋𝑛 < ∞
(2.14)
𝑛=1
𝑛𝛽𝑛 𝑋𝑛 < ∞.
(2.15)
Proof. Since 𝛽𝑛 → 0, then Δ𝛽𝑛 → 0, and hence
∞
∞
∞
∞
𝑣
∑
∑
∑
∑
∑
𝛽𝑛 𝑋𝑛 ≤
𝑋𝑛
∣ Δ𝛽𝑛 ∣=
∣ Δ𝛽𝑣 ∣
𝑋𝑛
𝑛=1
𝑣=𝑛
𝑛=1
=
∞
∑
∣ Δ𝛽𝑣 ∣
𝑣=1
= 𝑂(1)
𝑣=1
𝑣
∑
𝑛=1
𝑛𝛽 (log 𝑛)𝛾 𝑋𝑛 𝑛−𝛽 (log 𝑛)−𝛾
𝑛=1
∞
∑
∣ Δ𝛽𝑣 ∣ 𝑣 𝛽 (log 𝑣)𝛾 𝑋𝑣
𝑣=1
= 𝑂(1)
∞
∑
∣ Δ𝛽𝑣 ∣ 𝑣 𝛽 (log 𝑣)𝛾 𝑋𝑣
𝑣=1
= 𝑂(1)
∞
∑
𝑣
∑
𝑛=1
𝑣
∑
𝑛−𝛽 (log 𝑛)−𝛾
𝑛𝜖 (log 𝑛)−𝛾 𝑛−𝛽−𝜖 , 0 < 𝜖 < 𝛽 + 𝜖 < 1
𝑛=1
∣ Δ𝛽𝑣 ∣ 𝑣 𝛽 (log 𝑣)𝛾 𝑋𝑣 𝑣 𝜖 (log 𝑣)−𝛾
𝑣=1
= 𝑂(1)
∞
∑
𝑣=1
= 𝑂(1)
= 𝑂(1)
∞
∑
𝑣=1
∞
∑
𝑣=1
𝑣
∑
𝑛=1
∣ Δ𝛽𝑣 ∣ 𝑣 𝛽+𝜖 𝑋𝑣
∫
𝑣
𝑥−𝛽−𝜖 𝑑𝑥
0
∣ Δ𝛽𝑣 ∣ 𝑣 𝛽+𝜖 𝑋𝑣 𝑣 1−𝛽−𝜖
𝑣 ∣ Δ𝛽𝑣 ∣ 𝑋𝑣 = 𝑂(1),
𝑛−𝛽−𝜖
8
W. T. SULAIMAN
in view of (2.11).
∞
∑
𝑛𝛽𝑛 𝑋𝑛 = 𝑛𝑋𝑛
Δ𝛽𝑣 ≤ 𝑛𝑋𝑛
𝑛=𝑣
∞
∑
∣ Δ𝛽𝑣 ∣
𝑛=𝑣
= 𝑛1−𝛽 (log 𝑛)−𝛾 𝑛𝛽 (log 𝑛)𝛾 𝑋𝑛
∞
∑
∣ Δ𝛽𝑣 ∣
𝑛=𝑣
≤𝑛
1−𝛽
−𝛾
(log 𝑛)
∞
∑
𝑣 𝛽 (log 𝑣)𝛾 𝑋𝑣 ∣ Δ𝛽𝑣 ∣
𝑛=𝑣
≤
=
∞
∑
𝑛=𝑣
∞
∑
𝑣 1−𝛽 (log 𝑣)−𝛾 𝑋𝑣 𝑣 𝛽 (log 𝑣)𝛾 𝑋𝑣 ∣ Δ𝛽𝑣 ∣
𝑣𝑋𝑣 ∣ Δ𝛽𝑣 ∣< ∞,
𝑛=𝑣
in view of (2.11).
□
∑𝑛
Corollary 2.3. Let (𝑝𝑛 ) be a positive sequence such that 𝑃𝑛 = 𝑣=0 𝑝𝑣 → ∞, 𝑡𝑛
denote the 𝑛−th (𝐶, 1) mean of the sequence (𝑛𝑎𝑛 ), and let (𝜆𝑛 ) be a sequence of
numbers such that 𝑝𝑛 , 𝑡𝑛 , 𝜆𝑛 are all satisfying
𝑛𝑝𝑛 = 𝑂(𝑃𝑛 ), 𝑃𝑛 = 𝑂(𝑛𝑝𝑛 ),
∞
∑
∞
∑
𝑣 𝛿𝑘
𝑣=1
∞
∑
𝑣=1
𝑣 𝛿𝑘
𝑝𝑛
= 𝑂(𝑣 𝛿𝑘 /𝑃𝑣−1 ),
𝑃𝑛 𝑃𝑛−1
(2.17)
𝑝𝑣
∣ 𝑡𝑣 ∣𝑘 ∣ 𝜆𝑣 ∣𝑘 = 𝑂(1),
𝑃𝑣
(2.18)
𝑛𝛿𝑘
𝑛=𝑣+1
(
𝑃𝑣
𝑝𝑣
(2.16)
)𝑘−1
∣ 𝑡𝑣 ∣𝑘 ∣ Δ𝜆𝑣 ∣𝑘 = 𝑂(1),
(2.19)
∑
then the series
𝑎𝑛 𝜆𝑛 is summable ∣ 𝑅, 𝑝𝑛 ∣𝑘 , 𝑘 ≥ 1, 𝛿 ≥ 0.
Furthermore if (𝑋𝑛 ) is a quasi (𝛽−𝛾)−power increasing sequence, 0 < 𝛽 < 1, 𝛾 ≥ 0,
and if (𝛽𝑛 ) is a sequence of numbers satisfying (2.9)-(2.13), then the conditions
∑
(2.17) are (2.18) omitted in order to obtain the ∣ 𝑅, 𝑝𝑛 ∣𝑘 −summability of
𝑎𝑛 𝜆𝑛 .
Proof. The proof follows from Theorem 2.1 by putting 𝑇 ≡ (𝑅, 𝑝𝑛 ), that is
𝑡𝑛𝑣 =
𝑝𝑛 𝑃𝑣−1
𝑝𝑛 𝑝𝑣
𝑝𝑣 ˆ
, 𝑡𝑛𝑣 =
, Δ𝑣 ˆ
𝑡𝑛𝑣 = −
.
𝑃𝑛
𝑃𝑛 𝑃𝑛−1
𝑃𝑛 𝑃𝑛−1
□
References
[1] E. Savas, A study on absolute summability factors for a triangular matrix, Math. Ineq. Appl.
12 (2009) 41–46.
[2] W. T. Sulaiman, A study relation between two summability methods, Proc. Amer. Math. Soc.
115 (1992) 303–312.
A STUDY ON ABSOLUTE FACTORS FOR A TRIANGULAR MATRIX
9
Waad T. Sulaiman
Department of Computer Engineering , College of Engineering, University of Mosul,
Iraq
E-mail address: waadsulaiman@hotmail.com