Gen. Math. Notes, Vol. 23, No. 2, August 2014, pp.63-70
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
New Theorems for Absolute Matrix
Summability Factors
Hikmet Seyhan Özarslan1 and Enes Yavuz2
1
Department of Mathematics, Erciyes University, Kayseri, Turkey
E-mail: seyhan@erciyes.edu.tr
2
Department of Mathematics, Celal Bayar University, Manisa, Turkey
E-mail: enes.yavuz@cbu.edu.tr
(Received: 26-3-14 / Accepted: 21-5-14)
Abstract
In this paper, we have given two theorems for |A, pn ; δ|k summability which
generalize recent theorems on |A, pn |k summability. Study also reveals many
factor theorems for other summability methods.
Keywords: Absolute matrix summability, quasi power increasing sequences,
infinite series.
1
Introduction
A positive sequence (γn ) is said to be quasi β-power increasing sequence if
there exists a constant K = K(β, γ) ≥ 1 such that Knβ γn ≥ mβ γm holds for
all n ≥ m ≥ 1 (see [4]). A sequence (λn ) is said to be of bounded variation,
P
P
|∆λn | = ∞
denote by (λn ) ∈ BV, if ∞
n=1
n=1 |λn − λn+1 | < ∞.
P
Let
an be a given infinite series with the partial sums (sn ) and let
A = (anv ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping
the sequence s = (sn ) to As = (An (s)), where
An (s) =
n
X
v=0
anv sv ,
n = 0, 1, ...
(1)
64
The series
Hikmet Seyhan Özarslan et al.
P
an is said to be summable |A|k , k ≥ 1, if (see [9])
∞
X
¯ n (s)|k < ∞,
nk−1 |∆A
(2)
n=1
where
¯ n (s) = An (s) − An−1 (s).
∆A
Let (pn ) be a sequence of positive numbers such that
Pn =
n
X
pv → ∞ as n → ∞,
(P−i = p−i = 0, i ≥ 1).
(3)
v=0
The series
P
an is said to be summable |A, pn |k , k ≥ 1, if (see [8])
∞
X
Pn
pn
n=1
The series
P
!k−1
¯ n (s)|k < ∞,
|∆A
(4)
an is said to be summable |A, δ|k , k ≥ 1, if (see [7])
∞
X
¯ n (s)|k < ∞
nδk+k−1 |∆A
(5)
n=1
and it is said to be summable |A, pn ; δ|k , k ≥ 1 and δ ≥ 0, if (see [5])
∞
X
n=1
Pn
pn
!δk+k−1
¯ n (s)|k < ∞.
|∆A
(6)
In the special case when pn = 1, |A, pn ; δ|k summability is the same as |A, δ|k
summability. Also if we take δ = 0, then |A, pn ; δ|k summability is the
same as
pv
|A, pn |k summability. Finally, when anv = Pn the method reduces to N̄ , pn ; δ k
summability method (see [3]) and when anv = Ppvn , δ = 0 it reduces to N̄ , pn k
summability method (see [1]).
Now, we will introduce some further notations necessary for our main theorems.
Given a normal matrix A = (anv ), we associate two lower semimatrices Ā =
(ānv ) and  = (ânv ) as follows:
ānv =
n
X
ani ,
n, v = 0, 1, ...
(7)
i=v
and
â00 = ā00 = a00 ,
ânv = ānv − ān−1,v ,
n = 1, 2, ...
(8)
65
New Theorems for Absolute Matrix...
It may be noted that Ā and  are the well-known matrices of series-to-sequence
and series-to-series transformations, respectively. Then, we have
An (s) =
n
X
n
X
anv sv =
v=0
ānv av
(9)
v=0
and
n
X
¯ n (s) =
∆A
ânv av .
(10)
v=0
2
Main Result
In [6], Özarslan and Yavuz have proved two theorems for |A, pn |k summability
method by using quasi β-power increasing sequences. The aim of this paper
is to generalize their theorems to |A, pn ; δ|k summability. Now, we state our
main theorems.
Theorem 2.1 Let A = (anv ) be a positive normal matrix such that
ān0 = 1,
an−1,v ≥ anv ,
n = 0, 1, ...
m+1
X
n=v+1
m+1
X
n=v+1
Pn
pn
!δk
Pn
pn
n ≥ v + 1,
f or


Pv
|∆v ânv | = O
 pv
!δk
(12)
pn
,
=O
Pn
ann
(11)
(13)
!δk−1 



Pv
|ân,v+1 | = O
 pv
,
(14)

!δk 

(15)

and let there be sequences (βn ) and (λn ) such that
(λn ) ∈ BV,
(16)
|∆λn | ≤ βn ,
(17)
βn → 0 as
n → ∞,
(18)
66
Hikmet Seyhan Özarslan et al.
∞
X
n|∆βn |Xn < ∞,
(19)
n=1
|λn |Xn = O(1) as
n → ∞,
(20)
where (Xn ) is a quasi β-power increasing sequence for some 0 < β < 1.
If
n
X
v=1
m
X
n=1
then
P
Pn
pn
Pv
pv
!δk
|sv |k
= O(Xn ),
v
(21)
!δk−1
|sn |k = O(Xm ),
m → ∞,
(22)
an λn is summable |A, pn ; δ|k , k ≥ 1 and 0 ≤ δ < 1/k.
Theorem 2.2 Let conditions (11)–(20) and (22) of Theorem 2.1 be satisfied. If
∞
X
Pn |∆βn |Xn < ∞,
(23)
n=1
m
X
n=1
then
P
Pn
pn
!δk
|sn |k
= O(Xm ),
Pn
(24)
an λn is summable |A, pn ; δ|k , k ≥ 1 and 0 ≤ δ < 1/k.
We need following lemmas for the proof of our theorems.
Lemma 2.3 (see [4]). Let (Xn ) be a quasi β-power increasing sequence for
some 0 < β < 1. If conditions (18) and (19) are satisfied, then
nXn βn = O(1) as
∞
X
n → ∞,
Xn βn < ∞.
(25)
(26)
n=1
Lemma 2.4 Let (Xn ) be a quasi β-power increasing sequence for some
0 < β < 1. If conditions (18) and (23) are satisfied, then
Pn βn Xn = O(1),
∞
X
pn βn Xn < ∞.
(27)
(28)
n=1
The proof of Lemma 2.4 is similar to that of Bor in [2] and hence omitted.
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New Theorems for Absolute Matrix...
3
Proof of Theorem 2.1
Let (Tn ) denotes A-transform of the series
applying Abel’s transformation we have
¯ n =
∆T
=
=
=
=
n
X
v=1
n−1
X
v=1
n−1
X
v=1
n−1
X
v=1
n−1
X
P
an λn . Then, by (9), (10) and
ânv av λv
∆v (ânv λv )
v
X
ak + ânn λn
n
X
av
v=1
k=1
(ânv λv − ân,v+1 λv+1 ) sv + ann λn sn
(ânv λv − ân,v+1 λv+1 − ân,v+1 λv + ân,v+1 λv ) sv + ann λn sn
∆v (ânv )λv sv +
n−1
X
ân,v+1 ∆λv sv + ann λn sn
v=1
v=1
= Tn,1 + Tn,2 + Tn,3
say.
Since
|Tn,1 + Tn,2 + Tn,3 |k ≤ 3k |Tn,1 |k + Tn,2 |k + Tn,3 |k ,
to complete the proof of Theorem 2.1, it is sufficient to show that
∞
X
(Pn /pn )δk+k−1 |Tn,r |k < ∞,
f or
r = 1, 2, 3.
(29)
n=1
0
Firstly, applying Hölder’s inequality with indices k and k , where k > 1 and
1
+ k10 = 1, we get that
k
m+1
X
n=2
Pn
pn
!δk+k−1
k
|Tn,1 |
≤
m+1
X
Pn
pn
n=2
= O(1)
!δk+k−1
m+1
X
n=2
×
n−1
X
Pn
pn
n−1
X
!k
|∆v ânv ||λv ||sv |
v=1
!δk+k−1
n−1
X
!
k
k
|∆v ânv ||λv | |sv |
v=1
!k−1
|∆v ânv |
v=1
= O(1)
= O(1)
m+1
X
n=2
Pn
pn
m
X
k
v=1
!δk
n−1
X
!
|∆v ânv ||λv |k |sv |k
v=1
k
|λv | |sv |
m+1
X
n=v+1
Pn
pn
!δk
|∆v ânv |
68
Hikmet Seyhan Özarslan et al.
= O(1)
m
X
v=1
= O(1)
m
X
v=1
= O(1)
m−1
X
Pv
pv
!δk−1
Pn
pn
!δk−1
|λv |k−1 |λv ||sv |k
|λv ||sv |k
∆|λv |
v=1
v
X
r=1
+ O(1)|λm |
m
X
Pv
pv
v=1
= O(1)
m−1
X
Pr
pr
!δk−1
|sr |k
!δk−1
|sv |k
βv Xv + O(1)|λm |Xm
v=1
= O(1) as m → ∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.3. Since (λn ) ∈ BV
by (16), applying Hölder’s inequality with the same indices above, we have
m+1
X
n=2
Pn
pn
!δk+k−1
k
|Tn,2 |
≤
m+1
X
Pn
pn
n=2
= O(1)
!δk+k−1
m+1
X
n=2
×
n−1
X
Pn
pn
!k
n−1
X
|∆λv ||ân,v+1 ||sv |
v=1
!δk+k−1 n−1
X
!
k
|∆λv ||ân,v+1 ||sv |
v=1
!k−1
|∆λv ||ân,v+1 |
v=1
= O(1)
m+1
X
n=2
= O(1)
m
X
m
X
v=1
= O(1)
m
X
v=1
= O(1)
!δk
m−1
X
k
Pv
pv
!δk
∆(vβv )
m
X
v=1
v=1
βv |ân,v+1 ||sv |k ×
Pn
pn
!δk
|ân,v+1 |
|sv |k
(vβv )
v
v
X
Pv
pv
n−1
X
v=1
m+1
X
r=1
+ O(1)mβm
= O(1)
!
n=v+1
!δk
Pv
|sv |k βv
pv
v=1
m−1
X
n−1
X
v=1
βv |sv |
v=1
= O(1)
Pn
pn
Pr
pr
!δk
!δk
|sr |k
r
|sv |k
v
∆(vβv )Xv + O(1)mβm Xm
!k−1
|∆λv |
69
New Theorems for Absolute Matrix...
= O(1)
m−1
X
v|∆βv |Xv + O(1)
v=1
m−1
X
βv+1 Xv+1
v=1
+ O(1)mβm Xm
= O(1) as m → ∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.3.
Finally, by following the similar process as that in Tn,1 we have that
m
X
n=1
Pn
pn
!δk+k−1
k
|Tn,3 |
≤
m
X
n=1
= O(1)
Pn
pn
!δk+k−1
m
X
Pn
pn
n=1
|ann |k |λn |k |sn |k
!δk−1
|λn ||sn |k = O(1) as m → ∞.
So, we get
∞
X
(Pn /pn )δk+k−1 |Tn,r |k < ∞,
f or
r = 1, 2, 3.
n=1
This completes the proof of Theorem 2.1.
4
Proof of Theorem 2.2
Using Lemma 2.4 and proceeding as that in the proof of Theorem 2.1, replacPm
P
δk |sv |k
δk
k
ing m
(βv Pv ) we can easily prove
v=1 (Pv /pv )
v=1 (Pv /pv ) |sv | βv by
Pv
Theorem 2.2.
5
Conclusion
We have proved theorems dealing with |A, pn ; δ|k summability factors of infinite series. In these theorems, if we take pn = 1 then we have two new results
dealing with |A, δ|k summability factors of infinite series. Also, if we take
anv = Ppvn , then we have another two new results concerning |N̄ , pn ; δ|k summability. Finally, when (Xn ) is taken as almost increasing sequence, new factor
theorems for |A, pn ; δ|k summability are obtained.
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Hikmet Seyhan Özarslan et al.
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