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 ∑
𝑣=0
𝑡𝑛𝑣𝑠𝑣. (1.1)
A series∑𝑎𝑛 is said to be summable∣𝑇 ∣𝑘, 𝑘≥1,if
∞ ∑
𝑛=1
𝑛𝑘−1∣Δ𝑇𝑛−1∣𝑘<∞. (1.2)
One of the most interesting cases of∣𝑇 ∣𝑘-summability is the case when𝑇 is chosen
to be the Cesaro matrix. That is by putting
𝑡𝑛𝑣 = 𝐴
𝛼−1
𝑛−𝑣 𝐴𝛼
𝑛
, 𝑣= 0,1, ..., 𝑛, 𝐴𝛼𝑛 =
Γ(𝑛+𝛼+ 1)
Γ(𝛼+ 1)Γ(𝑛+ 1) , 𝑛= 0,1, ... .
Given any lower triangular matrix𝑇 one can associate the matrices𝑇 and𝑇 ,ˆ with
entries defined by
𝑡𝑛𝑣 =
𝑛 ∑
𝑖=𝑣
𝑡𝑛𝑖, 𝑛 𝑎𝑛𝑑 𝑖= 0,1,2, ..., ˆ𝑡𝑛𝑣 =𝑡𝑛𝑣−𝑡𝑛−1,𝑣,
2000Mathematics Subject Classification. 40F05, 40D25.
Key words and phrases. Absolute summability; summability factor; weight function. c
⃝2009 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted Jun, 2009. Published September, 2009.
respectively. With𝑠𝑛=∑𝑛𝑖=0𝑎𝑖𝜆𝑖, we define and derive the following
𝑡𝑛:= 𝑛 ∑
𝑣=0
𝑡𝑛𝑣𝑠𝑣= 𝑛 ∑
𝑣=0
𝑡𝑛𝑣 𝑣 ∑
𝑖=0
𝑎𝑖𝜆𝑖= 𝑛 ∑
𝑖=0
𝑎𝑖𝜆𝑖 𝑛 ∑
𝑣=𝑖 𝑡𝑛𝑣=
𝑛 ∑
𝑖=0
𝑡𝑛𝑖𝑎𝑖𝜆𝑖, (1.3)
𝑌𝑛 :=𝑡𝑛−𝑡𝑛−1=
𝑛 ∑
𝑖=0
𝑡𝑛𝑖𝑎𝑖𝜆𝑖−
𝑛−1
∑
𝑖=0
𝑡𝑛−1,𝑖𝑎𝑖𝜆𝑖=
𝑛 ∑
𝑖=0
ˆ
𝑡𝑛𝑖𝑎𝑖𝜆𝑖, 𝑎𝑠 𝑡𝑛−1,𝑛= 0. (1.4)
𝑋𝑛:=𝑢𝑛−𝑢𝑛−1=
𝑛 ∑
𝑖=0
ˆ
𝑢𝑛𝑖𝑎𝑖𝜇𝑖. (1.5)
We call 𝑇 a triangle if𝑇 is lower triangular and𝑡𝑛𝑛∕= 0 for all𝑛. A triangle𝐴is called factorable if its nonzero entries𝑎𝑚𝑛can be written in the form𝑏𝑚𝑐𝑛for each 𝑚 and𝑛. Recall thatˆ𝑡𝑛𝑛=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 constant𝐾=𝐾(𝛽, 𝛾)≥1 such that𝐾𝑛𝛽𝛾𝑛 ≥𝑚𝛽𝛾𝑚 holds for all𝑛≥𝑚≥1.
It should be mentioned that every almost increasing sequence is quasi𝛽−power in-creasing sequence for any𝛽 >0,while the converse need not be true as for example
𝛾𝑛=𝑛−𝛽, 𝛽 >0.
Here we generalize the quasi𝛽-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(log𝑛)𝛾𝜙
𝑛 ≥ 𝑚𝛽(log𝑚)𝛾𝜙𝑚 holds for all 𝑛≥𝑚≥1, 𝛾 ≥0.Clearly quasi(𝛽−0)−power increasing sequence is the quasi𝛽− power increasing sequence.
The series∑𝑎𝑛 is said to be summable∣𝑅, 𝑝𝑛∣
𝑘, 𝑘≥1, if ∞
∑
𝑛=1
𝑛𝑘−1∣𝜎𝑛−𝜎𝑛−1∣𝑘<∞,
where
𝜎𝑛=
1
𝑃𝑛 𝑛 ∑
𝑣=0
𝑝𝑣𝑠𝑣.
The (𝐶,1)−mean of the sequence (𝑓𝑛) is equal to 𝑛+11 ∑𝑛𝑣=0𝑓𝑣, and we said that
𝑓(𝑛) =𝑂(𝑔(𝑛)) if lim𝑛→∞𝑔𝑓((𝑛𝑛))=𝑘 <∞.
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), (iv)∑𝑛𝑣=1−1𝑎𝑣𝑣 ∣ˆ𝑎𝑛,𝑣+1∣=𝑂(𝑎𝑛𝑛). Let 𝑡1
𝑛 denote the𝑛−th(𝐶,1) mean of(𝑛𝑎𝑛).If (v)∑∞
𝑣=1𝑎𝑣𝑣∣𝜆𝑣∣
𝑘∣𝑡1
𝑣∣𝑘=𝑂(1), (vi)∑∞
𝑣=1∣𝑎𝑣𝑣∣
1−𝑘∣Δ𝜆𝑣∣𝑘∣𝑡1
𝑣∣𝑘=𝑂(1),
then the series∑
𝑎𝑛𝜆𝑛 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,
and𝑘is not restricted to integers but𝑘∈[1,∞).
3. An extension is made by assuming further hypothesis.
2. Results.
The following is our main result.
Theorem 2.1. Let𝑇 be a lower triangular matrix,𝑡𝑛 denote the𝑛−th(𝐶,1)mean of the sequence(𝑛𝑎𝑛), and let(𝜆𝑛)be a sequence of numbers such that𝑇, 𝑡𝑛, 𝜆𝑛 are all satisfying
𝑛∣𝑡𝑛𝑛∣=𝑂(1),1 =𝑂(𝑛∣𝑡𝑛𝑛∣), (2.1)
𝑛−1
∑
𝑣=1
∣𝑡𝑣𝑣 ∣∣ˆ𝑡𝑛𝑣 ∣=𝑂(∣𝑡𝑛𝑛∣), (2.2)
𝑛−1
∑
𝑣=1
∣𝑡𝑣𝑣∣∣ˆ𝑡𝑛,𝑣+1∣=𝑂(∣𝑡𝑛𝑛∣), (2.3)
∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘∣ˆ𝑡𝑛𝑣 ∣=𝑂(𝑣
𝛿𝑘), (2.4)
𝑛−1
∑
𝑣=1
∣Δ𝑣ˆ𝑡𝑛𝑣 ∣=𝑂(∣𝑡𝑛𝑛∣), (2.5)
∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘 ∣Δ𝑣ˆ𝑡𝑛𝑣 ∣=𝑂(𝑣𝛿𝑘∣𝑡𝑣𝑣 ∣), (2.6)
∞ ∑
𝑣=1
𝑣𝛿𝑘 ∣𝑡𝑣𝑣 ∣∣𝑡𝑣∣𝑘∣𝜆𝑣 ∣𝑘=𝑂(1), (2.7)
∞ ∑
𝑣=1
then the series∑
𝑎𝑛𝜆𝑛 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→0𝑎𝑠 𝑛→ ∞, (2.10)
∞ ∑
𝑛=1
𝑛∣Δ𝛽𝑛∣𝑋𝑛<∞, (2.11)
∣𝜆𝑛∣𝑋𝑛=𝑂(1), (2.12)
𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡𝑣 ∣𝑘
𝑋𝑣𝑘−1
=𝑂(𝑋𝑚), (2.13)
then the conditions (2.7) and (2.8) are omitted in order to obtain the ∣ 𝑇, 𝛿 ∣𝑘
−summability of∑ 𝑎𝑛𝜆𝑛.
Proof of Theorem 2.1. We have
𝑌𝑛:= 𝑛 ∑
𝑣=0
ˆ
𝑡𝑛𝑣𝜆𝑣𝑎𝑣= 𝑛 ∑
𝑣=1
𝑣𝑎𝑣 ˆ 𝑡𝑛𝑣𝜆𝑣
𝑣
=
𝑛−1
∑
𝑣=1
( 𝑣 ∑
𝑟=1
𝑟𝑎𝑟 )
Δ𝑣 (
ˆ 𝑡𝑛𝑣𝜆𝑣
𝑣 )
+
( 𝑛 ∑
𝑣=1
𝑣𝑎𝑣 )
(𝑡𝑛𝑛𝜆𝑛
𝑛 )
=
𝑛−1
∑
𝑣=1
(𝑣+ 1)𝑡𝑣 ( 1
𝑣(𝑣+ 1)ˆ𝑡𝑛𝑣𝜆𝑣+ 1
𝑣+ 1Δˆ𝑡𝑛𝑣𝜆𝑣+ 1
𝑣+ 1ˆ𝑡𝑛,𝑣+1Δ𝜆𝑣
)
+𝑛+ 1
𝑛 𝑡𝑛𝑡𝑛𝑛𝜆𝑛
=
𝑛−1
∑
𝑣=1
(1
𝑣𝑡𝑣ˆ𝑡𝑛𝑣𝜆𝑣+𝑡𝑣Δ𝑣ˆ𝑡𝑛𝑣𝜆𝑣+𝑡𝑣ˆ𝑡𝑛,𝑣+1Δ𝜆𝑣 )
+𝑛+ 1
𝑛 𝑡𝑛𝑡𝑛𝑛𝜆𝑛
=𝑌𝑛1+𝑌𝑛2+𝑌𝑛3+𝑌𝑛4.
In order to prove the Theorem, by Minkowski’s inequality, we have to show that
∞ ∑
𝑛=1
By Holder’s inequality
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣𝑌𝑛1∣𝑘= ∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣
𝑛−1
∑
𝑣=1 1
𝑣𝑡𝑣ˆ𝑡𝑛𝑣𝜆𝑣 ∣ 𝑘
≤
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1 𝑛−1
∑
𝑣=1
𝑣−𝑘 ∣𝑡𝑣𝑣 ∣1−𝑘∣𝑡𝑣 ∣𝑘∣ˆ𝑡𝑛𝑣∣∣𝜆𝑣∣𝑘 (𝑛−1
∑
𝑣=1
∣𝑡𝑣𝑣∣∣ˆ𝑡𝑛𝑣 ∣ )𝑘−1
=𝑂(1)
∞ ∑
𝑛=2
𝑛𝛿𝑘(𝑛∣𝑡𝑛𝑛∣)𝑘−1
𝑛−1
∑
𝑣=1
𝑣−𝑘 ∣𝑡𝑣𝑣 ∣1−𝑘∣𝑡𝑣∣𝑘∣
ˆ𝑡𝑛𝑣 ∣∣𝜆𝑣∣ 𝑘
=𝑂(1)
∞ ∑
𝑣=1
𝑣−𝑘∣𝑡𝑣𝑣∣1−𝑘∣𝑡𝑣∣𝑘∣𝜆𝑣∣𝑘 ∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘∣ˆ𝑡𝑛𝑣 ∣
=𝑂(1)
∞ ∑
𝑣=1
𝑣𝛿𝑘−𝑘 ∣𝑡𝑣𝑣 ∣1−𝑘∣𝑡𝑣 ∣𝑘∣𝜆𝑣∣𝑘
=𝑂(1)
∞ ∑
𝑣=1
𝑣𝛿𝑘 ∣𝑡𝑣𝑣 ∣∣𝑡𝑣∣𝑘∣𝜆𝑣∣𝑘=𝑂(1),
in view of (2.2) and (2.7).
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣𝑌𝑛2∣𝑘= ∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣
𝑛−1
∑
𝑣=1
𝑡𝑣Δ𝑣ˆ𝑡𝑛𝑣𝜆𝑣 ∣𝑘
≤
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1 𝑛−1
∑
𝑣=1
∣𝑡𝑣∣𝑘∣Δ𝑣ˆ𝑡𝑛𝑣 ∣∣𝜆𝑣∣𝑘 (𝑛−1
∑
𝑣=1
∣Δ𝑣ˆ𝑡𝑛𝑣∣ )𝑘−1
=𝑂(1)
∞ ∑
𝑛=2
𝑛𝛿𝑘(𝑛∣𝑡𝑛𝑛∣)𝑘−1
𝑛−1
∑
𝑣=1
∣𝑡𝑣 ∣𝑘∣Δ
𝑣ˆ𝑡𝑛𝑣 ∣∣𝜆𝑣∣𝑘
=𝑂(1)
∞ ∑
𝑣=1
∣𝑡𝑣∣𝑘∣𝜆𝑣∣𝑘 ∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘∣Δ𝑣ˆ𝑡𝑛𝑣∣
=𝑂(1)
∞ ∑
𝑣=1
in view of (2.5), (2.6) and (2.7).
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣𝑌𝑛3∣𝑘= ∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣
𝑛−1
∑
𝑣=1
𝑡𝑣ˆ𝑡𝑛,𝑣+1Δ𝜆𝑣∣𝑘
≤
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1 𝑛−1
∑
𝑣=1
∣𝑡𝑣 ∣𝑘∣𝑡𝑣𝑣∣1−𝑘∣ˆ𝑡𝑛,𝑣+1∣∣Δ𝜆𝑣∣𝑘 (𝑛−1
∑
𝑣=1
∣𝑡𝑣𝑣 ∣∣ˆ𝑡𝑛,𝑣+1∣
)𝑘−1
=𝑂(1)
∞ ∑
𝑛=2
𝑛𝛿𝑘(𝑛∣𝑡𝑛𝑛∣)𝑘−1 𝑛−1
∑
𝑣=1
∣𝑡𝑣∣𝑘∣𝑡𝑣𝑣 ∣1−𝑘∣ˆ𝑡𝑛,𝑣+1∣∣Δ𝜆𝑣∣𝑘
=𝑂(1)
∞ ∑
𝑣=1
∣𝑡𝑣∣𝑘∣𝑡𝑣𝑣 ∣1−𝑘∣Δ𝜆𝑣∣𝑘 ∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘∣ˆ𝑡𝑛,𝑣+1∣
=𝑂(1)
∞ ∑
𝑣=1
𝑣𝛿𝑘 ∣𝑡𝑣 ∣𝑘∣𝑡𝑣𝑣∣1−𝑘∣Δ𝜆𝑣∣𝑘=𝑂(1),
in view of (2.3), (2.4) and (2.8).
∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣𝑌𝑛4∣𝑘= ∞ ∑
𝑛=2
𝑛𝛿𝑘+𝑘−1∣ 𝑛+ 1
𝑛 𝑡𝑛𝑡𝑛𝑛𝜆𝑛∣ 𝑘
=𝑂(1)
∞ ∑
𝑛=1
𝑛𝛿𝑘+𝑘−1∣𝑡𝑛 ∣𝑘∣𝑡𝑛𝑛∣𝑘∣𝜆𝑛∣𝑘
=𝑂(1)
∞ ∑
𝑛=1
𝑛𝛿𝑘 ∣𝑡𝑛𝑛∣∣𝑡𝑛∣𝑘∣𝜆𝑛∣𝑘=𝑂(1),
in view of (2.7).
In order to complete the proof the rest, it is sufficient to show that
∞ ∑
𝑣=1
𝑣𝛿𝑘∣𝑡𝑣𝑣 ∣∣𝑡𝑣 ∣𝑘∣𝜆𝑣 ∣𝑘=𝑂(1),
∞ ∑
𝑣=1
𝑣𝛿𝑘 ∣𝑡𝑣 ∣𝑘∣𝑡𝑣𝑣 ∣1−𝑘∣Δ𝜆𝑣∣𝑘=𝑂(1).
Now,
𝑚 ∑
𝑣=1
𝑣𝛿𝑘∣𝑡𝑣𝑣∣∣𝑡𝑣 ∣𝑘∣𝜆𝑣∣𝑘=𝑂(1) 𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡
𝑣∣𝑘 𝑋𝑣𝑘−1
𝑋𝑣𝑘−1∣𝜆𝑣∣𝑘−1∣𝜆𝑣∣
=𝑂(1)
𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡𝑣∣𝑘
𝑋𝑣𝑘−1
∣𝜆𝑣∣
=𝑂(1)
𝑚−1
∑
𝑣=1
( 𝑣 ∑
𝑟=1
𝑟𝛿𝑘−1∣𝑡𝑟∣𝑘
𝑋𝑟𝑘−1 )
∣Δ∣𝜆𝑣∣ ∣+𝑂(1)
(𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡𝑣 ∣𝑘
𝑋𝑣𝑘−1 )
∣𝜆𝑚∣
=𝑂(1)
𝑚−1
∑
𝑣=1
𝑋𝑣∣Δ𝜆𝑣∣+𝑂(1)𝑋𝑚∣𝜆𝑚∣
=𝑂(1)
𝑚−1
∑
𝑣=1
𝑚 ∑
𝑣=1
𝑣𝛿𝑘∣𝑡𝑣∣𝑘∣𝑡𝑣𝑣 ∣1−𝑘∣Δ𝜆𝑣∣𝑘=𝑂(1) 𝑚 ∑
𝑣=1
𝑣𝛿𝑘+𝑘−1∣𝑡𝑣∣𝑘∣Δ𝜆𝑣 ∣𝑘
=𝑂(1)
𝑚 ∑
𝑣=1
𝑣𝛿𝑘∣𝑡 𝑣∣𝑘 𝑋𝑣𝑘−1
(𝑣𝑋𝑣𝛽𝑣)𝑘−1𝛽𝑣
=𝑂(1)
𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡𝑣∣𝑘
𝑋𝑣𝑘−1 𝑣𝛽𝑣
=𝑂(1)
𝑚−1
∑
𝑣=1
( 𝑣 ∑
𝑟=1
𝑟𝛿𝑘−1∣𝑡𝑟∣𝑘
𝑋𝑟𝑘−1 )
∣Δ(𝑣𝛽𝑣)∣+𝑂(1)
(𝑚 ∑
𝑣=1
𝑣𝛿𝑘−1∣𝑡𝑣∣𝑘
𝑋𝑣𝑘−1 )
𝑚𝛽𝑚
=𝑂(1)
𝑚 ∑
𝑣=1
𝑋𝑣𝛽𝑣+𝑂(1) 𝑚 ∑
𝑣=1
𝑣𝑋𝑣 ∣Δ𝛽𝑣∣+𝑂(1)𝑚𝑋𝑚𝛽𝑚
=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 ∞
∑
𝑛=1
𝛽𝑛𝑋𝑛<∞ (2.14)
𝑛𝛽𝑛𝑋𝑛 <∞. (2.15)
Proof. Since𝛽𝑛 →0,then Δ𝛽𝑛 →0,and hence ∞ ∑ 𝑛=1 𝛽𝑛𝑋𝑛≤ ∞ ∑ 𝑛=1 𝑋𝑛 ∞ ∑ 𝑣=𝑛 ∣Δ𝛽𝑛 ∣= ∞ ∑ 𝑣=1 ∣Δ𝛽𝑣 ∣ 𝑣 ∑ 𝑛=1 𝑋𝑛 = ∞ ∑ 𝑣=1 ∣Δ𝛽𝑣∣ 𝑣 ∑ 𝑛=1
𝑛𝛽(log𝑛)𝛾𝑋𝑛𝑛−𝛽(log𝑛)−𝛾
=𝑂(1)
∞ ∑
𝑣=1
∣Δ𝛽𝑣 ∣𝑣𝛽(log𝑣)𝛾𝑋𝑣 𝑣 ∑
𝑛=1
𝑛−𝛽(log𝑛)−𝛾
=𝑂(1)
∞ ∑
𝑣=1
∣Δ𝛽𝑣 ∣𝑣𝛽(log𝑣)𝛾𝑋𝑣 𝑣 ∑
𝑛=1
𝑛𝜖(log𝑛)−𝛾𝑛−𝛽−𝜖, 0< 𝜖 < 𝛽+𝜖 <1
=𝑂(1)
∞ ∑
𝑣=1
∣Δ𝛽𝑣 ∣𝑣𝛽(log𝑣)𝛾𝑋𝑣𝑣𝜖(log𝑣)−𝛾
𝑣 ∑
𝑛=1
𝑛−𝛽−𝜖
=𝑂(1)
∞ ∑ 𝑣=1 ∣Δ𝛽𝑣 ∣𝑣𝛽+𝜖𝑋𝑣 ∫ 𝑣 0 𝑥−𝛽−𝜖𝑑𝑥
=𝑂(1)
∞ ∑
𝑣=1
∣Δ𝛽𝑣 ∣𝑣𝛽+𝜖𝑋𝑣𝑣1−𝛽−𝜖
=𝑂(1)
∞ ∑
𝑣=1
in view of (2.11).
𝑛𝛽𝑛𝑋𝑛 =𝑛𝑋𝑛 ∞ ∑
𝑛=𝑣
Δ𝛽𝑣≤𝑛𝑋𝑛 ∞ ∑
𝑛=𝑣
∣Δ𝛽𝑣∣
=𝑛1−𝛽(log𝑛)−𝛾𝑛𝛽(log𝑛)𝛾𝑋𝑛 ∞ ∑
𝑛=𝑣
∣Δ𝛽𝑣 ∣
≤𝑛1−𝛽(log𝑛)−𝛾
∞ ∑
𝑛=𝑣
𝑣𝛽(log𝑣)𝛾𝑋𝑣∣Δ𝛽𝑣∣
≤
∞ ∑
𝑛=𝑣
𝑣1−𝛽(log𝑣)−𝛾𝑋𝑣𝑣𝛽(log𝑣)𝛾𝑋𝑣∣Δ𝛽𝑣∣
=
∞ ∑
𝑛=𝑣
𝑣𝑋𝑣∣Δ𝛽𝑣 ∣<∞,
in view of (2.11). □
Corollary 2.3. Let (𝑝𝑛) be a positive sequence such that 𝑃𝑛 =∑𝑛𝑣=0𝑝𝑣 → ∞, 𝑡𝑛
denote the 𝑛−th (𝐶,1) mean of the sequence (𝑛𝑎𝑛), and let (𝜆𝑛) be a sequence of numbers such that𝑝𝑛, 𝑡𝑛, 𝜆𝑛 are all satisfying
𝑛𝑝𝑛 =𝑂(𝑃𝑛), 𝑃𝑛=𝑂(𝑛𝑝𝑛), (2.16)
∞ ∑
𝑛=𝑣+1
𝑛𝛿𝑘 𝑝𝑛 𝑃𝑛𝑃𝑛−1
=𝑂(𝑣𝛿𝑘/𝑃𝑣−1), (2.17)
∞ ∑
𝑣=1
𝑣𝛿𝑘𝑝𝑣 𝑃𝑣 ∣𝑡𝑣∣
𝑘∣𝜆𝑣∣𝑘=𝑂(1), (2.18)
∞ ∑
𝑣=1
𝑣𝛿𝑘 (𝑃
𝑣 𝑝𝑣
)𝑘−1
∣𝑡𝑣 ∣𝑘∣Δ𝜆𝑣∣𝑘=𝑂(1), (2.19)
then the series∑𝑎
𝑛𝜆𝑛 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∑𝑎𝑛𝜆𝑛.
Proof. The proof follows from Theorem 2.1 by putting𝑇 ≡(𝑅, 𝑝𝑛),that is
𝑡𝑛𝑣 = 𝑝𝑣 𝑃𝑛
, ˆ𝑡𝑛𝑣 =
𝑝𝑛𝑃𝑣−1
𝑃𝑛𝑃𝑛−1
, Δ𝑣ˆ𝑡𝑛𝑣=− 𝑝𝑛𝑝𝑣 𝑃𝑛𝑃𝑛−1
.
□
References
[1] E. Savas,A study on absolute summability factors for a triangular matrix, Math. Ineq. Appl. 12(2009) 41–46.
Waad T. Sulaiman
Department of Computer Engineering , College of Engineering, University of Mosul, Iraq
