Internat. J. Math. & Math. Sci.
Vol. 22, No. 4 (1999) 817–822
S 0161-17129922817-7
© Electronic Publishing House
CHARACTERIZATION ON SOME ABSOLUTE SUMMABILITY
FACTORS OF INFINITE SERIES
W. T. SULAIMAN
(Received 12 June 1998)
Abstract. A general theorem concerning some absolute summability factors of infinite
series is proved. This theorem characterizes as well as generalizes our previous result [4].
Other results are also deduced.
Keywords and phrases. Summability, series, sequences.
1991 Mathematics Subject Classification. 40D15, 40F15, 40F05.
1. Introduction. Let an be an infinite series with partial sum sn . Let σnδ and ηδn
denote the nth Cesàro mean of order δ(δ > −1) of the sequences {sn } and {nan },
respectively. The series an is said to be summable |C, δ|k , k ≥ 1, if
∞
n=1
k
δ < ∞,
nk−1 σnδ − σn−1
(1.1)
or, equivalently,
∞
k
n−1 ηn < ∞.
(1.2)
n=1
Let {pn } be a sequence of positive real constants such that
Pn =
n
pv → ∞ as n → ∞.
(1.3)
v=0
The series
an is said to be summable N, pn k , k ≥ 1, if (Bor [1])
k−1
∞
Pn
Tn − Tn−1 k < ∞,
pn
n=1
(1.4)
where
Tn = Pn−1
n
pv sv .
(1.5)
v=0
For pn = 1, N, pn k summability is equivalent to C, 1k summability. In general,
the two summabilities are not comparable. Let {ϕn } be any sequence of positive real
constants. The series an is said to be summable N, pn , ϕn k , k ≥ 1, if (Sulaiman [4])
∞
n=1
k
k−1 ϕn
Tn − Tn−1 < ∞.
(1.6)
818
W. T. SULAIMAN
Clearly,
N, pn , Pn = N, pn ,
k
pn k
N, 1, n = C, 1 .
k
k
(1.7)
Theorem 1.1 (Sulaiman [4]). Let {pn }, {qn }, and {ϕn } be sequences of real positive
constants. Let tn denote the (N, pn )-mean of the series an . If
∞
n=1
k Pn
pn
∞
n=1
∞
n=1
Pn
pn
qn
Qn
k
k k
k−1 ϕn
n ∆tn−1 < ∞,
k k
k−1 ϕn
n ∆tn−1 < ∞,
k
(1.8)
k k
k−1 ϕn
∆n ∆tn−1 < ∞,
then the series an n is summable N, qn , ϕn k , k ≥ 1, where ∆fn = fn −fn+1 for any
sequence {fn } and
Qn =
n
qv → ∞
as n → ∞
q−1 = Q−1 = 0 .
(1.9)
v=0
2. Lemmas
Lemma 2.1 (Bor [1]). Let k > 1 and A = (anv ) be an infinite matrix. In order that
A ∈ (k ; k ), it is necessary that
anv = O(1)
(all n, v).
(2.1)
Lemma 2.2. Suppose that n = O(fn gn ), fn , gn ≥ 0, {n /fn gn } monotonic, ∆gn =
O(1), and ∆fn = O(fn /gn+1 ). Then ∆n = O(fn ).
Proof. Let kn = (n /fn gn ) = O(1). If (kn ) is nondecreasing, then
∆n = kn fn gn − kn+1 fn+1 gn+1
≤ kn fn gn − kn fn+1 gn+1
= kn ∆ fn gn = kn fn ∆gn + gn+1 ∆fn ,
∆n = O fn ∆gn + O gn+1 ∆fn = O f n + O fn = O f n .
(2.2)
If (kn ) is nonincreasing, write ∇fn = fn+1 − fn ,
∇n = kn+1 fn+1 gn+1 − kn fn gn
≤ kn ∇ fn gn
= kn fn ∇gn + gn+1 ∇fn ,
∆n = ∇n = O fn ∇gn + O gn+1 ∇fn = O fn ∆gn + O gn+1 ∆fn = O f n + O fn = O f n .
(2.3)
CHARACTERIZATION ON SOME ABSOLUTE SUMMABILITY FACTORS . . .
819
3. Main Result. We state and prove the following theorem:
Theorem 3.1. Let {pn }, {qn }, {αn }, and {βn } be sequences of positive real numbers
such that
βn qn
is nonincreasing;
(3.1)
Qn
pn Qn = O Pn qn ;
(3.2)

1−(1/k) 
P q

βn
n n
n is monotonic;
(3.3)
 pn Qn αn

Qn
= O(1);
(3.4)
∆
qn


1−(1/k) 
1−(1/k) 
p

p q

αn
αn
n
n n+1
∆
=O
.
(3.5)
 P n βn

 Pn Qn+1 βn

Then the necessary and sufficient
conditions
that an n be summable N, qn , βn k ,
whenever an is summable N, pn , αn k , k ≥ 1, are
pn Qn αn 1−(1/k)
,
(3.6)
n = O
Pn qn βn

1−(1/k) 
 p

αn
n
.
(3.7)
∆n =
 Pn−1 βn

Proof. Write
n
qn
Qv−1 av v ,
Qn Qn−1 v=1
n
pn
1−(1/k)
tn = αn
Pv−1 av ,
Pn Pn−1 v=1
1−(1/k)
Tn = βn
Tn =
1−(1/k)
βn
1−(1/k)
= βn
1−(1/k)
= βn
qn
Qn Qn−1
qn
Qn Qn−1
qn
Qn Qn−1
n
Pv−1 av
v=1
n−1 v
(3.8)
Qv−1
v
Pv−1
n
Qn−1
Qv−1
n
+
Pr −1 ar
Pv−1 v
Pn−1
r =1
pv Qv v Qv
−qv
α(1/k)−1
t
+
+
∆
v
v
v
v
Pv−1
Pv−1 Pv
Pv
Pr −1 ar ∆
v=1 r =1
n−1
Pv Pv−1
v=1
pv
Pn Pn−1
qn
Qn−1
tn α(1/k)−1
n
n
Qn Qn−1
pn
Pn−1
n−1
−qv
qn
1−(1/k)
= βn
Pv α(1/k)−1
tv v
v
Qn Qn−1 v=1 pv
Pv Qv (1/k)−1
(1/k)−1
+ αv
Qv tv v +
αv
tv ∆v
pv
Pn qn (1/k)−1 1−(1/k)
+
α
βn
tn n .
pn Qn n
1−(1/k)
+ βn
Let us denote the above form of Tn by Tn,1 + Tn,2 + Tn,3 + Tn,4 .
(3.9)
820
W. T. SULAIMAN
By Minkowski’s inequality, in order to prove the sufficiency, it is sufficient to show
∞
that n=1 |Tn,r |k < ∞, r = 1, 2, 3, 4. Applying Hölder’s inequality,
m+1
n=2
|Tn,1 |k =
m+1
k
n−1
−qv
1−(1/k)
qn
β
Pv α(1/k)−1
t
v
v
n
v
Qn Qn−1 v=1 pv
n=2

k−1
k
k
n−1
n−1
qv 
q
P
1
k
k
n
v
k−1
≤
βn
qv
αv1−k tv v 
Qn
Qn−1 v=1
pv
Qn−1 
n=2
v=1
k
k
m
k k m+1
1
Pv
qn
k−1
≤ O(1)
qv
αv1−k tv v βn
p
Q
Q
v
n
n−1
v=1
n=v+1
k
k−1 m+1
m
k
Pv
qv
qn
1−k k = O(1)
qv
αv
tv
v
βv
p
Q
Q
Qn−1
v
v
n
v=1
n=v+1
k
k−1
m
k k
βv
qv Pv
tv v ,
= O(1)
p
Q
α
v
v
v
v=1
m+1
k
n−1
m+1
1−(1/k)
qn
β
Tn,2 k =
α(1/k)−1
Qv tv v n
v
Q
Q
n n−1 v=1
n=2
n=2
m+1
m+1
k
1
n−1
Qv
qv
k

k−1
qv 
k k n−1
qv tv v 
Qn−1 
α1−k
Qn−1 v=1 v
v=1
k
k
m
m+1
k k
Qv
qn
1
k−1
≤ O(1)
αv1−k
qv tv v βn
q
Q
Q
v
n
n−1
v=1
n=v+1
k
k−1 m+1
m
k k
qv
qn
1−k Qv
= O(1)
αv
qv tv
v
βv
q
Q
Q
Qn−1
v
v
n
v=1
n=v+1
m
k−1
βv
k k
tv v = O(1)
α
v
v=1
k k k−1
m
k k
Pv
qv
βv
tv v ,
= O(1)
p
Q
α
v
v
v
v=1
=
n=2
k−1
βn
qn
Qn
k
n−1
m+1
Pv−1
1−(1/k)
qn
(1/k)−1
β
Tn,3 k =
Q
α
t
∆
v v
v
v
n
Q
Q
p
n
n−1
v
n=2
n=2
v=1
k
k
k
n−1
m+1
qn
Pv−1
Qv
1 k−1
≤
βn
qv
αv1−k
Qn
Qn−1 v=1
pv
qv
n=2

k−1
qv 
k k n−1
× tv
∆v

Qn−1 
v=1
k
k
k
m
k m+1
Pv−1
Qv
qn
1
1−k k k−1
= O(1)
qv
αv
βn
tv
∆v
pv
qv
Qn
Qn−1
v=1
n=v+1
k k
k−1
m
k
Qv
qv Pv−1
qv
1−k k tv
∆v
= O(1)
αv
βv
Qv
pv
qv
Qv
v=1
m+1
CHARACTERIZATION ON SOME ABSOLUTE SUMMABILITY FACTORS . . .
= O(1)
m
v=1
m+1
m
Tn,4 k = O(1)
n=2
n=1
Pv−1
pv
k qn Pn
pn Qn
βv
αv
k k−1
βn
αn
821
k tv ∆v k ,
k−1
k k
tn n .
(3.10)
Sufficiency of (3.6) and (3.7) follows.
Necessity of (3.6). Using the result of Bor in [2], the transformation from (tn )
into (Tn ) maps k into k and, hence by Lemma 2.1 the diagonal elements of this
transformation are bounded and so (3.6) is necessary.
Necessity of (3.7). This follows from Lemma 2.2 and the necessity of (3.6) by
taking
fn =
pn
Pn
αn
βn
1−(1/k)
,
gn =
Qn
.
qn
(3.11)
4. Applications
Corollary 4.1. Suppose that the conditions (3.1) and (3.2) are satisfied. Then the
necessary and sufficient condition that an be summable |N, qn , βn |k , whenever it is
summable |N, pn , αn |k , k ≥ 1, is


 α 1−(1/k) 
P n qn
n
=O
.
(4.1)
 βn

pn Qn
Proof. The proof follows from Theorem 3.1 by putting n = 1 and noticing that
we do not need the conditions (3.3), (3.4), and (3.5) as ∆n = 0 for n = 1.
Corollary 4.2. Suppose that (3.2) and (3.4) are satisfied, {(Pn qn /pn Qn )(1/k) n } is
monotonic, and


1−(1/k) 
1−(1/k) 
p

p q

Pn qn
Pn qn
n
n n+1
∆
=O
.
(4.2)
 Pn pn Qn

 Pn Qn+1 pn Qn

Then the necessary and sufficient conditions that an n be summable |N, qn |k when
ever an is summable |N, pn |k , k ≥ 1, are

1/k
1−(1/k) 

 p
pn Qn
Pn qn
n
n = O
.
(4.3)
,
∆n =


Pn q n
Pn−1 pn Qn
Proof. The proof follows from Theorem 3.1 by putting αn = Pn /pn , βn = Qn /qn .
Corollary 4.3 (Bor and Thorpe [3]). Suppose that pn Qn = O(Pn qn ) and Pn qn =
O(pn Qn ). Then, the series an is summable |N, qn |k if and only if it is summable
|N, pn |k , k ≥ 1.
Proof. The proof follows from the sufficient part of Corollary 4.1.
822
W. T. SULAIMAN
Remark. It may be noticed that (3.4) can be replaced by
Qn ∆qn = O qn qn+1 ,
as
Qn Qn Qn+1 qn+1 Qn − qn (Qn + qn+1 ) ∆
=
=
−
qn qn
qn+1 qn qn+1
Q ∆q
n n
=
+ 1
qn qn+1
≤ 1+
(4.4)
(4.5)
Qn |∆qn |
.
qn qn+1
References
[1]
[2]
[3]
[4]
H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1,
147–149. MR 86d:40004. Zbl 554.40008.
, On the relative strength of two absolute summability methods, Proc. Amer. Math.
Soc. 113 (1991), no. 4, 1009–1012. MR 92c:40006. Zbl 743.40007.
H. Bor and B. Thorpe, On some absolute summability methods, Analysis 7 (1987), no. 2,
145–152. MR 88j:40012. Zbl 639.40005.
W. T. Sulaiman, On some summability factors of infinite series, Proc. Amer. Math. Soc. 115
(1992), no. 2, 313–317. MR 92i:40009. Zbl 756.40006.
Sulaiman: College of Education, Ajman University, P. O. Box 346, Ajman, United Arab
Emirates