Download On the Generalization of a Theorem of Bor - PMF-a

Filomat 31:6 (2017), 1595–1600
DOI 10.2298/FIL1706595A
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
On the Generalization of a Theorem of Bor
Tuba Arıa
a P.
O. Box 23, TR-38002 Kayseri, Turkey
Abstract. In [6], Bor proved a theorem dealing with absolute Riesz summability factors of infinite series.
In this paper, we generalize that result for the absolute matrix summability factors of infinite series. Some
new results are also obtained.
1. Introduction
A positive sequence (bn ) is said to be an almost increasing sequence if there exists a positive increasing
sequence (cn ) and two positive constants A and B suchPthat Acn ≤ P
bn ≤ Bcn (see [1]). A sequence (λn ) is said
to be of bounded variation, denoted by (λn ) ∈ BV, if n |∆λn | = n | λn − λn+1 |< ∞. A positive sequence
X = (Xn ) is said to be a quasi-f-power increasing sequence if there exists a constant K = K(X, f ) ≥ 1 such
that K fn Xn ≥ fm Xm for all n ≥ m ≥ 1, where f = ( fn ) = {nδ (log n)σ , σ ≥ P
0, 0 < δ < 1} (see [12]). If we take
σ = 0, then we get a quasi-δ-power increasing sequence (see [10]). Let an be a given infinite series with
partial sums
P (sn ). We denote by (un ) and (tn ) the nth (C, 1) means of the sequence (sn ) and (nan ), respectively.
The series an is said to be summable |C, 1|k , k ≥ 1, if (see [7], [9])
∞
X
k−1
n
n=1
∞
X
1 k
|un − un−1 | =
|tn | < ∞.
n
k
(1)
n=1
Let (pn ) be a sequence of positive numbers such that
Pn =
n
X
pv → ∞ as
n → ∞, (P−i = p−i = 0, i ≥ 1).
(2)
v=0
The sequence-to-sequence transformation
vn =
n
1 X
pv sv
Pn v=0
(3)
2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99, 46A45
Keywords. Sequence space, Riesz mean, absolute matrix summability, power increasing sequences, infinite series, Hölder inequality,
Minkowski inequality
Received: 06 March 2015; Accepted: 02 May 2015
Communicated by Dragan S. Djordjević
Email address: [email protected] (Tuba Arı)
T. Arı / Filomat 31:6 (2017), 1595–1600
1596
defines the sequence (vn ) of the Riesz mean or simply
P the (N̄, pn ) mean of the sequence (sn ), generated by
the sequence of coefficients (pn ) (see [8]). The series an is said to be summable | N̄, pn |k , k ≥ 1, if (see [2])
∞
X
(Pn /pn )k−1 | vn − vn−1 |k < ∞.
(4)
n=1
In the special case pn = 1 for all values of n | N̄, pn |k summability is the same as | C, 1 |k summability.
b = (b
Given a normal matrix A = (anv ), we associate two lower semi-matrices A = (anv ) and A
anv ) as follows:
anv =
n
X
ani , n, v = 0, 1, ...
(5)
i=v
and
b
anv = anv − an−1,v , n = 1, 2, ...
a00 = a00 = a00 ,b
(6)
b are the well-known matrices of series-to-sequence and series-to-series transIt may be noted that A and A
formations, respectively. Then, we have
An (s) =
n
X
anv sv , n = 0, 1, ...
(7)
v=0
and
An (s) − An−1 (s) =
n
X
b
anv av .
(8)
v=0
The series
∞
X
P
an is said to be summable |A, pn |k , k ≥ 1, if (see [11])
(Pn /pn )k−1 | ∆An (s) |k < ∞,
(9)
n=1
where
∆An (s) = An (s) − An−1 (s).
In the special case, for anv = pv /Pn , |A, pn |k summability is the same as | N̄, pn |k summability.
2. The Known Result
Quite recently, Bor has proved the following theorem dealing with | N̄, pn |k summability factors of
infinite series.
Theorem 2.1 ([6]). Let (λn ) ∈ BV and let (Xn ) be a quasi-f-power increasing sequence for some δ (0 < δ < 1)
and σ ≥ 0. Suppose that there exists sequences (βn ) and (λn ) such that
| ∆λn |≤ βn ,
βn → 0
∞
X
as
(10)
n → ∞,
n | ∆βn | Xn < ∞,
(11)
(12)
n=1
| λn | Xn = O(1).
(13)
T. Arı / Filomat 31:6 (2017), 1595–1600
1597
If
n
X
| tv |k
= O(Xn ) as
v
n → ∞,
(14)
v=1
and (pn ) is a sequence such that
Pn = O(npn ),
(15)
Pn ∆pn = O(pn pn+1 ),
(16)
Pn λ n
n=1 an npn
P∞
are satisfied, then the series
is summable | N̄, pn |k , k ≥ 1.
It should be noted that if we take σ = 0, then we get a result which was proved in [4].
3. The Main Result
The aim of this paper is to generalize Theorem 2.1 by using absolute matrix summability factors. Now,
we shall prove the following theorem.
Theorem 3.1. Let A = (anv ) be a positive normal matrix such that
ān0 = 1, n = 0, 1, 2, ...,
(17)
an−1,v ≥ anv , f or n ≥ v + 1
p n
ann = O
Pn
nann = O(1)
ân,v+1 = O (v|∆v ânv |) .
(18)
(19)
(20)
(21)
Let (λn ) ∈ BV and let (Xn ) be a quasi-f-power increasing sequence for some δ (0 < δ < 1) and σ ≥ 0. If the
P
Pn λ n
conditions (10)-(16) are satisfied, then the series ∞
n=1 an npn is summable | A, pn |k , k ≥ 1.
p
It should be noted that if we take anv = Pvn , then we get Theorem 2.1.
We require the following lemmas for the proof of our theorem.
Lemma 3.2 ([3]). If the conditions (15) and (16) are satisfied, then ∆ nP2 pnn = O n12 .
Lemma 3.3 ([5]). Except for the condition (λn ) ∈ BV under the conditions on (Xn ), (βn ) and (λn ) as expressed
in the statement of the theorem, we have the following;
nXn βn = O(1),
∞
X
(22)
βn Xn < ∞.
(23)
n=1
Proof of Theorem 3.1. Let (Tn ) be the A-transform of the series
Tn − Tn−1 =
n
X
ânv
v=1
P∞
n=1
an Pn λn
npn .
av Pv vλv
.
v2 pv
Using Abel’s transformation, we get that
! v
n−1
n
X
Pv λv X
ânn Pn λn X
∆v ânv 2
Tn − Tn−1 =
rar +
vav
v pv r=1
n2 pn v=1
v=1
n−1
X
n−1
X
Pv
λv
tv Pv
ân,v+1 (v + 1) 2 ∆λv
(v + 1) 2 tv +
pv
pv
v
v
v=1
v=1
!
n−1
X
Pv
ann Pn λn
+
ân,v+1 λv+1 (v + 1)tv ∆ 2
+
(n + 1)tn
v
p
n2 pn
v
v=1
=
=
∆v (ânv )
Tn,1 + Tn,2 + Tn,3 + Tn,4
Then, we have
T. Arı / Filomat 31:6 (2017), 1595–1600
1598
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
!k−1
∞
X
Pn
| Tn,r |k < ∞,
pn
f or
r = 1, 2, 3, 4.
(24)
n=1
When k > 1, we can apply Hölder’s inequality with indices k and k0 , where
!k−1
m+1
X
Pn
| Tn,1 |k
p
n
n=2
1
k
+
1
k0
= 1 and so we get that
k
!k−1 
m+1
n−1


X


Pn
X

= O(1)
|∆v (ânv )||λv ||tv |





p
n
n=2
v=1

  n−1
k−1
!k−1 X
m+1
X
 n−1
 X

Pn

|∆v ânv ||λv |k ||tv |k  
= O(1)
|∆v ânv |

pn
n=2
v=1
v=1
!k−1
m+1
n−1
X
X
Pn
(ann )k−1
|∆v ânv ||λv |k |tv |k
= O(1)
p
n
n=2
v=1
= O(1)
= O(1)
m
X
v=1
m
X
m+1
X
|λv |k |tv |k
|∆v ânv |
n=v+1
|λv ||tv |k avv = O(1)
v=1
= O(1)
m−1
X
v=1
= O(1)
m−1
X
m
X
|λv ||tv |k
v=1
pv
Pv
m
v
X
X
pv k
pr k
|tr | + O(1)|λm |
|tv |
∆|λv |
Pr
Pv
v=1
r=1
|∆λv |Xv + O(1) | λm | Xm
v=1
= O(1)
m−1
X
βv Xv + O(1) | λm | Xm = O(1)
v=1
as m → ∞, by virtue of the hypotheses of Theorem 3.1 and Lemma 3.3.
Now, by using (15), we have that
!k−1
m+1
X
Pn
| Tn,2 |k
p
n
n=2
=
=
=
=
k
!k−1 
m+1
n−1


X


Pn
X

O(1)
| ân,v+1 || ∆λv || tv |





p
n
n=2
v=1


k
!k−1 X
m+1
n−1

X


Pn


O(1)
v|∆v ânv ||∆λv ||tv |





pn
n=2
v=1
  n−1
k−1
!k−1 X
m+1
X
 n−1
 X

Pn
k
k
k



 (v|∆v ânv |) (βv ) |tv |  
|∆v ânv |
O(1)



pn
n=2
v=1
v=1
!
m
m+1
k−1
X
X P
n
O(1)
(vβv )(vβv )k−1 |tv |k
(ann )k−1 |∆v ânv |
pn
v=1
= O(1)
m
X
n=v+1
vβv |tv |k avv (vβv )k−1
v=1
= O(1)
m
X
v=1
vk−1 vβv
m
X
1
| tv |k
k
|
t
|
=
O(1)
vβv
v
k
v
v
v=1
T. Arı / Filomat 31:6 (2017), 1595–1600
=
O(1)
m−1
X
∆(vβv )
v=1
=
O(1)
m−1
X
1599
v
m
X
X
| tr |k
| tv |k
+ O(1)mβm
r
v
r=1
v=1
| (v + 1)∆βv − βv | Xv + O(1)mβm Xm
v=1
=
O(1)
m−1
X
v | ∆βv | Xv + O(1)
v=1
m−1
X
βv Xv + O(1)mβm Xm = O(1), as m → ∞,
v=1
by virtue of hypotheses
of Theorem 3.1 and Lemma 3.3.
Now, since ∆ vP2 pv v = O v12 , by Lemma 3.2, we have
!k−1
m+1
X
Pn
| Tn,3 |k
p
n
n=2
k
!k−1 
m+1
n−1

X


Pn
1

X
= O(1)
|ân,v+1 ||λv+1 ||tv | 





p
v
n
n=2
v=1


 n−1
k−1
!k−1 X
m+1
X
 n−1
 X

Pn
k k 


 
= O(1)
|∆
â
||λ
|
|t
|
|∆
â
|
v
nv
v+1
v
nv

v


pn
n=2
v=1
v=1
!k−1
n−1
m+1
X
X
Pn
k−1
(ann )
|∆v ânv ||λv+1 |k |tv |k
= O(1)
p
n
n=2
v=1
= O(1)
= O(1)
m
X
v=1
m
X
m+1
X
|λv+1 |k |tv |k
|∆v ânv |
n=v+1
|λv+1 |k |tv |k avv
v=1
= O(1)
m−1
X
v=1
= O(1)
= O(1)
m−1
X
v=1
m
X
∆|λv+1 |
m
v
X
X
|tv |k
|tr |k
+ O(1)|λm+1 |
r
v
v=1
r=1
∆|λv |Xv + O(1)|λm+1 |Xm
βv Xv + O(1)|λm+1 |Xm+1 = O(1)
v=1
as m → ∞ by (10), (13), (14), (15), (20) and (21).
Finally, as in Tn,3 , we have
!k−1
m
m
X
X
1
Pn
| Tn,4 |k = O(1)
nk−1 k | λn |k−1 | λn || tn |k
pn
n
n=1
n=1
= O(1)
m
X
| tn | k
= O(1),
n
as
m → ∞.
n=1
p
This completes the proof of Theorem 3.1. If we take anv = Pvn and pn = 1 for all values of n, then we obtain a
new result concerning the | C, 1 |k summability factors of infinite series.
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