PORTUGALIAE MATHEMATICA
Vol. 56 Fasc. 3 – 1999
CHARACTERIZATION FOR RELATIONS
ON SOME SUMMABILITY METHODS
W.T. Sulaiman
Abstract: In this paper we characterize a previous result proved by us connecting
the summability methods |N , pn |k with either |N, qn |k or |N , wn |k for given sequences
{pn }, {qn } and {wn } of positive real constants. Other results are also deduced.
1 – Introduction
Let
an be an infinite series with partial sums sn . Let σnδ and ηnδ denote
the n-th Cesàro mean of order δ (δ > −1) of the sequences {sn } and {n an }
P
respectively. The series an is said to be summable |C, δ|k , k ≥ 1, if
P
∞
X
δ
nk−1 |σnδ − σn−1
|k < ∞ ,
n=1
or equivalently
∞
X
1
n
n=1
|ηnδ |k < ∞ .
Let {pn } be a sequence of real or complex constants such that
Pn = p 0 + p 1 + · · · + p n
The series
P
(p−1 = P−1 = 0) .
an is said to be summable |N, pn | if
(1.1)
∞
X
n=1
Received : October 19, 1997.
|Tn − Tn−1 | < ∞ ,
274
W.T. SULAIMAN
where
Tn =
n
1 X
pn−v sv
Pn v=0
(T−1 = 0) .
We write p = {pn } and
n
o
M = p : pn > 0 and pn+1 /pn ≤ pn+2 /pn+1 ≤ 1, n = 0, 1, ... .
It is known for p ∈ M (1.1) holds iff (see [5])
n
¯
1 ¯¯ X
pn−v v av ¯¯ < ∞ .
¯
n Pn v=1
n=1
∞
X
For p ∈ M , we say that
P
¯
¯
an is summable |N, pn |k , k ≥ 1, if
¯
¯k
∞
n
X
¯
1 ¯¯ 1 X
¯
p
v
a
n−v
v¯ < ∞
¯
n P
n=1
(Sulaiman [6]) .
n v=1
In the special case in which pn = Anr−1 , r > −1, where Arn is the coefficient of
xn in the power series expansion of (1 − x)−r−1 for |x| < 1, |N, pn |k reduces to
P
|C, r|k summability. The series an is said to be summable |N , pn |k , k ≥ 1, if
¶
∞ µ
X
Pn k−1
n=1
pn
where
|tn − tn−1 |k < ∞
(Bor [1]) ,
n
1 X
tn =
pv s v .
Pn v=0
If we take pn = 1, then |N , pn |k summability is equivalent to |C, 1|k summability.
In general these two summabilities are not comparable.
Throughout this paper we set
Qn = q 0 + q 1 + · · · + q n ,
q−1 = Q−1 = 0 ,
Wn = w 0 + w 1 + · · · + w n ,
w−1 = W−1 = 0 ,
∆fn = fn − fn+1 .
Let {pn } and {qn } be sequences of positive real constants such that q ∈ M .
is said to be summable |N, pn , qn |k , k ≥ 1, if
n
¯X
¯k
pn
¯
¯
P
q
a
v−1 n−v v ¯ < ∞
¯
k
P
R
n=1 n n−1 v=1
∞
X
¯
¯
(Sulaiman [7]) ,
P
an
RELATIONS ON SOME SUMMABILITY METHODS
275
where
Rn = p0 qn + p1 qn−1 + · · · + pn q0 .
Clearly |N, pn , 1|k is the same as |N , pn |k .
The following results are known.
Theorem A (Bor [1]).
such that as n → ∞
(1.2)
If
P
Let {pn } be a sequence of positive real constants
(i)
n pn = O(Pn ) ,
(ii)
Pn = O(n pn ) .
an is summable |C, 1|k , then it is summable |N , Pn |k , k ≥ 1.
Theorem B (Bor [2]). Let {pn } be a sequence of positive real constants
P
such that it satisfies (1.2). If an is summable |N , pn |k then it is also summable
|C, 1|k .
Theorem C (Sulaiman [7]). Let {pn }, {qn } and {wn } be sequences of
k
positive real constants such that q ∈ M and {pn /Pn Rn−1
} is nonincreasing for
P
qn 6= c. Let tn denote the (N , wn )-mean of the series an . Let {εn } be a sequence
of constants. If
m
X
pn qn−v−1
(1.3)
n=v+1
¶
∞ µ
X
Pn k−1
(1.4)
n=1
(1.5)
P
n=1 n
= O(Pv−1 ) ,
n=1
wn
|εn |k |∆tn−1 |k < ∞ ,
pn
wn
|∆εn |k |∆tn−1 |k < ∞ ,
µ
¶ µ
¶
∞
X
pn Pn−1 k Wn k
P Rn−1
n=1 n
P
m→∞,
|εn |k |∆tn−1 |k < ∞ ,
¶
µ
¶
∞ µ
X
Pn k−1 Wn k
and
then the series
pn
µ
¶
∞
X
pn W n k
(1.6)
(1.7)
Pn Rn−1
wn
|εn |k |∆tn−1 |k < ∞ ,
an is summable |N, pn , qn |k , k ≥ 1.
It may be mentioned that Theorems A and B are special cases of Theorem C.
The object of this paper is to prove the following
276
W.T. SULAIMAN
Theorem D. Let (pn ), (qn ) and (wn ) be sequences of positive real constants
k
such that q ∈ M and (pn /Pn Rn−1
) nonincreasing for qn 6= c. Suppose that
(1.8)
Rn−1 = O(Pn−1 ) ,
(1.9)
Pn wn = O(pn Wn ) ,
wn Pn Rn−1
∆
Wn pn Pn−1
µ
(1.10)
∆
(1.11)
µ
¶
wn
=O
Wn
µ
¶
,
Wn pn Pn−1
pn Pn−1
εn = O
wn Pn Rn−1
Pn Rn−1
¶
µ
¶
.
Then necessary and sufficient conditions that
an εn be summable |N, pn , qn |k
P
whenever an is summable |N , wn |k , k ≥ 1, are
P
wn Pn Rn−1
εn = O
Wn pn Pn−1
µ
(i)
wn
∆εn = O
Wn−1
µ
(ii)
¶
¶
,
.
2 – Lemmas
Lemma 1 (Sulaiman [7]). Let q ∈ M . Then for 0 < γ ≤ 1,
∞
X
qn−v−1
= O(v −γ ) .
γQ
n
n−1
n=v+1
Lemma 2 (Bor [4]). Let k ≥ 1 and let A = (anv ) be an infinite matrix. In
order that A ∈ (`k ; `k ) it is necessary that
(2.1)
anv = O(1)
(for all n, v) .
Lemma 3. Suppose that εn = O(fn gn ), fn , gn ≥ 0, fn+1 gn+1 = O(fn gn ),
∆(fn gn ) = O(fn ) and ∆(εn /fn gn ) = O(1/gn ). Then ∆εn = O(fn ).
Proof: We have
εn = k n f n g n ,
where kn =
εn
= O(1) ,
fn g n
∆εn = kn fn+1 ∆gn + kn gn+1 ∆fn + fn+1 gn+1 ∆kn .
RELATIONS ON SOME SUMMABILITY METHODS
277
Since
fn ∆gn + gn+1 ∆fn = O(fn ) ,
then
h
i
∆εn = kn fn ∆gn + kn O(fn ) − fn ∆gn + fn+1 gn+1 ∆kn
= kn O(fn ) + O(fn gn |∆kn |)
= O(fn ) + O(fn )
= O(fn ) .
3 – Proof of Theorem D
Write
Tn =
n
X
Pv−1 qn−v av εv ,
tn =
v=1
Tn =
n
X
Wv−1 av
v=1
=
n−1
v
X µX
=
n−1
X
v=1
(
Pv−1
qn−v εv
Wv−1
¶
Wr−1 ar ∆v
v=1 r=1
(3.1)
µ
Pv−1 ∆v qn−v
+ pv qn−v−1
Wv−1
wv
µ
n
X
wn
Wv−1 av ,
Wn Wn−1 v=1
¶
¶ µX
¶
n
Pv−1
Pn−1
qn−v εv +
Wr−1 ar
q 0 εn
Wv−1
Wn−1
r=1
Wv
εv tv + Pv−1 qn−v−1 εv tv
wv
)
Wv−1
Wn
εv tv − Pv qn−v−1
∆εv tv + Pn−1 q0
εn t n
wv
wn
= Tn,1 + Tn,2 + Tn,3 + Tn,4 + Tn,5 ,
say .
In order to prove sufficiency, by Minkowski’s inequality, it is sufficient to show
that
∞
X
pn
|Tn,r |k < ∞ ,
k
P
R
n=1 n n−1
r = 1, 2, 3, 4, 5 .
278
W.T. SULAIMAN
Applying Hölder’s inequality,
m+1
X
n=2
m+1
X
X
¯n−1
¯k
pn
pn
Wv
k
¯
¯
ε
t
|T
|
=
P
∆
q
v v¯
n,1
v−1 v n−v
¯
k
wv
Pn Rn−1
P Rk
n=1 n n−1 v=1
¯
m+1
X
≤
µ
¶
n−1
X
Wv k
pn
k
P
|∆
q
|
|εv |k |tv |k
v n−v
v−1
k
w
Pn Rn−1
v
v=1
n=1
½n−1
X
·
|∆v qn−v |
v=1
= O(1)
= O(1)
m+1
X
n=2
m
X
v=1
m
X
k
Pv−1
pv
P
v=1 v
µ
m+1
X
pn
pn
k
|T
|
=
n,4
k
Pn Rn−1
P Rk
n=2 n n−1
m+1
X
≤
¯
pn
Pn Rn−1
n=2
½n−1
X
·
v=1
= O(1)
= O(1)
¾k−1
µ
Wv
wv
Pv−1
Rv−1
¶k
v=1
n−1
Xµ
v=1
¶
m µ
X
Pv k
n=v
¶k µ
Wv
wv
¶k
|εv |k |tv |k ,
v
Pv
pv
¶k
pv qn−v−1
µ
Wv−1
wv
¶k
|∆εv |k |tv |k
¾k−1
pv
µ
Wv−1
wv
¶k
pv
v=1
µ
m
X Pv ¶k−1 µ Wv−1 ¶k
v=1
pn |∆v qn−v |
k
Pn Rn−1
¯k
¯n−1
¯
¯X
Wv−1
¯
∆εv tv ¯¯
Pv qn−v−1
¯
w
pv qn−v−1
Rn−1
pv
m+1
X
|εv |k |tv |k
wv
|∆εv |k |tv |k
m+1
X
pn qn−v−1
Pn Rn−1
n=v+1
|∆εv |k |tv |k .
Similarly we can show that
m+1
X
¶
m µ
X
pn
Pv k−1
k
|T
|
=
O(1)
|εv |k |tv |k ,
n,2
k
p
Pn Rn−1
v
v=1
m+1
X
µ
¶
m
X
pv W v k
pn
k
|T
|
=
O(1)
|εv |k |tv |k ,
n,3
k
P
w
Pn Rn−1
v
v=1 v
n=2
n=2
µ
¶ µ
¶
m
X
pn Pn−1 k Wn k
pn
k
|Tn,5 | = O(1)
|εn |k |tn |k .
k
P
R
w
P
R
n
n−1
n
n
n−1
n=1
n=1
m
X
The sufficiency follows.
RELATIONS ON SOME SUMMABILITY METHODS
279
Necessity of (i). Using the result of Bor in [4], the transformation from
((Pn /pn )1−1/k tn ) into ([(pn /Pn )1/k /Rn−1 ] Tn ) maps `k into `k and hence the diagonal elements of this transformation are bounded (by Lemma 2) and so (i) is
necessary.
Necessity of (ii). This follows from Lemma 3 and the necessity of (i) by taking
fn = wn /Wn , gn = Pn Rn−1 /pn Pn−1 .
4 – Applications
Corollary 1. Let {pn } and {wn } be sequences of positive real constants
such that (1.9) is satisfied.
P
Then the necessary and sufficient conditions such that
an be summable
|N , pn |k whenever it is summable |N , wn |k , k ≥ 1, is
(4.1)
pn Wn = O(Pn wn ) .
The proof follows from Theorem D by putting εn = 1, qn = 1.
Corollary 2 (Bor and Thorpe [3]). Let {pn } and {wn } be sequences of
positive real constants such that (1.9) and (4.1) are satisfied.
P
Then the series an is summable |N , pn |k iff it is summable |N , wn |k , k ≥ 1.
The proof follows from Corollary 1.
Corollary 3. Let (pn ), (wn ) be sequences of positive real constants such
that (1.9) is satisfied and
wn P n
∆
W n pn
µ
¶
wn
=O
Wn
µ
pn
W n pn
εn = O
∆
wn P n
Pn
¶
µ
µ
Then necessary and sufficient conditions that
P
whenever an is summable |N , wn |k , k ≥ 1, are
wn P n
,
εn = O
W n pn
µ
¶
¶
¶
P
,
.
an εn be summable |N , pn |k
wn
∆εn = O
Wn
µ
¶
.
The proof follows from Theorem D by putting qn = εn = 1.
280
W.T. SULAIMAN
Remark. It may be mentioned that Theorems A and B could be obtained
from Corollary 2.
REFERENCES
[1] Bor, H. – On two summability methods, Math. Proc. Cambridge Philos. Soc., 97
(1985), 147–149.
[2] Bor, H. – A note on two summability methods, Proc. Amer. Math. Soc., 98 (1986),
81–84.
[3] Bor, H. and Thorpe, B. – On some absolute summability methods, Analysis, 7
(1987), 145–152.
[4] Bor, H. – On the relative strength of two absolute summability methods, Proc.
Amer. Math. Soc., 113 (1991), 1009–1012.
[5] Das, G. – Tauberian theorems for absolute Nörlund summability, Proc. London
Math. Soc., 19 (1969), 357–384.
[6] Sulaiman, W.T. – Notes on two summability methods, Pure Appl. Math. Sci.,
31(1990), 59–68.
[7] Sulaiman, W.T. – Relations on some summability methods, Proc. Amer. Math.
Soc., 118 (1993), 1139–1145.
W.T. Sulaiman,
P.O. Box 120054,
Doher – QATAR