86 SOME NOTES ON ABSOLUTE SUMMABILITY FACTORS
*W. T. Sulaiman
Department of Computer Engineering, College of Engineering, University of Mosul, Iraq
*Author for Correspondence ABSTRACT
In this paper we give improvement to the result of [3] concerning absolute summability of infinite series.
2000 (MSC): 40F05, 40D25.
Key Words: Absolute Summability, Summability Factor INTRODUCTION
Let T be a lower triangular matrix,
s
n a sequence, and : .0
n
v v nv
n t s
T (1.1) A series
an is said to be summable T , k 1 ,
k if
1 .
1
1
n k nk T
n (1.2)
Given any lower triangular matrix T one can associate the matrices
T
and Tˆ, with entries defined bynv nv n v
n
v i
ni
nv t n i t t t
t , , 0,1,2..., ˆ 1,
respectively. With
,
0
ni i i
n
a
s
.
0 0
0 0
0
n
i
i i ni n
i v
nv n
i i i n
v v
i i i nv n
v v nv
n t s t a a t t a
t
(1.3): ˆ , 1, 0.
0 1
0 , 1 0
1
n nn
i
i i ni n
i
i i i n n
i
i i ni n
n
n t t t a t a t a ast
Y
(1.4)
We call T a triangle if T is lower triangular and
t
nn 0
for all n.We assume that
p
n is a sequence of positive real numbers such thatP
n p
0 p
1 ... p
n as n .
In the special case whent
nv p
v/ P
n,
summabilityT
k reduces ton k
p
N , summability.
Rhoads and Savas (2005) proved the following result
Theorem 1.1. Let A be a triangle with nonnegative entries satisfying (i)
a
n0 1 , n 0 , 1 , ...,
(ii) an1,v anv for nv1, (iii)
na
nn ( 1 ), 1 ( na
nn) ,
87
(iv)
ˆ ( ) .
0 , 1 nn
n
v
a
vva
nv a
If
X
n
is a positive monotonic nondecreasing sequence such that (v)
nX
n ( 1 ) ,
(vi) 2 (1),
1
n mn nXn
(vii)
mn1a
nnt
n k X
m ,
where, 1
1
1
n
k k
n
ka
t n
then the series
an
n is summable A , k 1 .
k
The object of this paper is to give two improvements to theorem 1.1 as follows 1. By weakening the condition (v) ,
2. By weakening the condition (vii), Main Result
In what follows we prove the following
Theorem 2.1. Let A be a triangle with nonnegative entries satisfying (i)
a
n0 1 , n 0 , 1 , ...,
(ii) an1,v anv for nv1, (iii)
na
nn ( 1 ), 1 ( na
nn) ,
(iv)
ˆ ( ) .
0 , nn
n
v
a
vva
nv a
(v) 1
ˆ
, 1( 1 ) .
1
1
mn v nv kk
a
n
If
X
n
is a positive monotonic nondecreasing sequence and if
n is a sequence of constants satisfying (vi)
n ( 1 ) ,
(vii) 2 (1),
1
n mn nXn
(viii) mn 1 k 1
m ,
n k n
nn
X
X t
a
where,
1 1
1
n
k k
n
ka
t n
then the series
an
n is summable A , k 1 .
k
It may be mentioned as well that whenever
X
n ,
condition (viii) of theorem 2.1 is weaker than condition (vii) of theorem 1.1. For if (viii) of theorem 1.1 is satisfied, thenX
n
implies that, 0
n
while if
n 0 ,
then by choosing, 0 , , (1/2)
2 /
1
n Xn n
nwe obtain
nX
n ( n
) ( 1 ) .
88
Although we add conditions (v), but this condition with (vi) caused for no losing powers via the proof.As an example in the proof of theorem 1.1,
n k1 has been replaced by ( 1 ),
and this is not the case in theorem 2.1.
Lemma 2.2. Condition (viii) of theorem 2.1 is weaker than (vii) of theorem 1.1 Proof. If (viii) holds, then we have
m
k n m
n k nn m
n k n
k n
nn a t X
X X
t
a
1 1 1 1
1
1 ,
while if (vii) is satisfied then,
1
1 1 1
nk
m
n
k k n n nn m
n
k n
nn
t X
X t a
a
1
1 1 1
1
1
1 1
mk
m
n k n
k n nn m
n
k n n
v k v
k v
vv
X
X t X a
X t a
1
11
1
1
nk m mkm
n
n X X X
X
mk
m
n
k n k n
m X X X
X
1
1
1 1
1 1
mk
k k
m
m X X X
X
1 1 11
( m),k
m X
X
for k 1.
Lemma 2.3. Conditions (vi)-(vii) of theorem 2.1 implies
nX
n
n ( 1 ) ,
(2.1) ,1
n
n
Xn
(2.2) and (2.2) implies
nX
n ( 1 ) .
(2.3) Proof. Since
n 0 ,
then
n 0 ,
and hence
(1)
2 (1)
2 (1).
v n
v v n
v
v n
n v
v n
n
n nX nX vX
nX
m
m
n n n
m
n n
v v m
n
n
n
X X
X
1
1
1 1
1
89
(1) (1) (1)1
1
2
m m m
n
n
n mX
nX
As
n 0 ,
(1)
(1).
v n
v v n
v v n
n
n X X
X
Lemma 2.4. Under the conditions of theorem 1.2, 1 ˆ
,0
nn n
v
nv
va a
(2.4)
1
1
) ˆ (
m
v n
vv nv
va a . (2.5)
ˆ ( 1 ) .
1
1 1
,
m
v n
v
a
n (2.6) For the proof see [3].Proof of Theorem 1.2.
Let
Y
n denote the nth term of A-transform of the series an
n , then
n
v
v nv v n
v
v v nv
n
v
a a v a
a Y
1 0
ˆ ˆ
:
n
a a v v
a a
r
nn nn
v v v
nv v n
v v
r r
1 1
1 1
ˆ
nv v v nv v nv v n nn n
n
v
v
t a
n a n
a v a v
v t v
v 1
1 ˆ ˆ 1
1 ˆ 1
) 1 ( ) 1 1
(
, 11
1
n nn n
n
v
v v n v v nv v v v nv
v t a
n a n
t a t a
vt ˆ
ˆ
ˆ
1
1 1
1
1 ,
Y
n1 Y
n2 Y
n3 Y
n4.
In order to prove the Theorem, by Minkowski's inequality, we have to show that
, 1,2,3,4.
1
1
Y j
n nj k
n k
By Holder's inequality,
n k
v
v nv v m
n k k n m
n
k
t a
n v Y
n
1
1 2
1 1
2
1
1 ˆ
90
1 1
1 1
1
1 2
1
ˆ ˆ
n k
v
nv vv k
v nv k v k vv n
v k m
n
k
v a t a a a
n
nv v kk v k vv n
v k m
n
k
nn v a t a
na ˆ
) 1
( 1
1
1 2
1
1 1
) ˆ 1 (
v n
nv k
v k v k vv m
v
vv va t a
a
v k v k
m
v vv t
a
1
) 1 (
11
)
11
(
v v v km
v k v
k v
vv
X
X t
a
v
m
v k v
k v vv
X t
a
1
)
11 (
1
1 1
1
( 1 )
) 1 (
m
v
m v
r k r
k r rr
v
X
t
a
m
v k v
k v vv
X t a
1 1
v m m
m
v
v
X X
1
1
) 1 (
( 1 ) .
in view of (iv), (2.2) ,(2.3) and (2.6) .
n k
v
v nv v v n
k k n n
k
Y n t a
n
1
1 2
1 2
2
1
ˆ
1 1
1 1
1 2
1
ˆ ˆ
n k
v
nv v k
v nv v n
v k v n
k
t a a
n
v nv v kn
v k v n
k
nn t a
na ˆ
) 1 (
1
1 2
1
v nv
v n k v v
k
v a
t ˆ
) 1 (
1 1
v k
v
k v vvt
a
1
) 1 (
( 1 ) ,
as in the case ofY
n1.
n k
v
v v n v m
n k k n m
n
k
Y n t a
n
1
1 1 , 2
1 3
2
1
ˆ
91
1 1
1 1
1
1 , 1
2
1
ˆ
k
v n
v v v
n
v
k v n k v k v m
n
k
t X a X
n
v
n
v
k v n k v k v m
n
k t X a
n
1
1
1 , 1
2
1 ˆ
) 1 (
m
v n
k v n k v
m
v
k v k
v X n t
t
1
1 , 1
1
1 ˆ
) 1
(
v
m
v k v
k v
vv v
X t
a
1
) 1
1 (
m
v k v
k v vv m v
r k r
k r rr m
v
v
X
t m a
X t v a
1 1 1
1 1
1
) 1 ( )
1
(
m m
m
v
v v m
v
v
v
v X m X
X
) 1 ( )
1 ( )
1 (
1
1
2 1
1
( 1 ) .
in view of (2.1), (2.2), (v), and (viii) .
k
n nn n n
k k n n
k t a
n n n Y
n 1
2 1 4
2
1
n k nnk nk
n
k t a
n
1
) 1
1 (
(1) (1),
1
k n k n n
nnt
a
as in the case of
Y
n2
This completes the proof of the Theorem .
Corollary 2.5. Let
(i)
np
n ( P
n) , P
n ( np
n) ,
(ii)
1/
.1 1
1 k
v k
n n
n v
n
k P
P P
n p
If
X
n
is a positive nondecreasing sequence and the sequence
n is satisfying conditions (vi)-(vii) of theorem 2.1, and
m
m
n k
n n
k n
n
X
X P
t
p
1 1 , (2.7)92
then the series
an
n is summable N,p , k1.
n k
REFERENCES
[1] H. Bor (1993). On absolute summability factors. Proceedings of the American Mathematical Society. 118. 71-75.
[2] T. M. Flett (1957). On an extension of absolute summability and some theorems of Littlewood and Paley. Proceedings of the London Mathematical Society. Third Series 7.
113-141.
[3] B. E. Rhoads and E. Savas (2005). A note on absolute summability factors. Periodica Mathematica Hungarica, 51 (1). 53-60.