International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 66-70
http://www.etamaths.com

ON GENERALIZED ABSOLUTE MATRIX SUMMABILITY METHODS

HİKMET SEYHAN ÖZARSLAN∗

Abstract. In this paper, we prove a general theorem dealing with absolute matrix summability
methods of infinite series. This theorem also includes some new and known results.

1. Introduction

Let
∑
an be a given infinite series with the partial sums (sn). Let (pn) be a sequence of positive

numbers such that

Pn =

n∑
v=0

pv →∞ as n →∞, (P−i = p−i = 0, i ≥ 1) .(1)

The sequence-to-sequence transformation

σn =
1

Pn

n∑
v=0

pvsv(2)

defines the sequence (σn) of the
(
N̄,pn

)
mean of the sequence (sn), generated by the sequence of

coefficients (pn) (see [5]). The series
∑
an is said to be summable

∣∣N̄,pn∣∣k ,k ≥ 1, if (see [1])
∞∑

n=1

(
Pn
pn

)k−1
|σn −σn−1|

k
< ∞.(3)

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∑
v=0

anvsv, n = 0, 1, ...(4)

The series
∑
an is said to be summable |A,pn|k ,k ≥ 1, if (see [6])

∞∑
n=1

(
Pn
pn

)k−1 ∣∣∆̄An(s)∣∣k < ∞,(5)
where

∆̄An(s) = An(s) −An−1(s).
Let (ϕn) be any sequence of positive real numbers. The series

∑
an is summable ϕ−|A,pn|k, k ≥ 1,

if
∞∑

n=1

ϕk−1n |∆̄An(s)|
k < ∞.(6)

If we take ϕn =
Pn
pn

, then ϕ−|A,pn|k summability reduces to |A,pn|k summability. If we set ϕn = n
for all n, ϕ−|A,pn|k summability is the same as |A|k summability (see [7]). Also, if we take ϕn =

Pn
pn

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99.
Key words and phrases. Riesz mean; summability factors; absolute matrix summability; infinite series, Hölder in-

equality; Minkowski inequality.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

66



ON GENERALIZED ABSOLUTE MATRIX SUMMABILITY METHODS 67

and anv =
pv
Pn

, then we get |N̄,pn|k summability. If we take ϕn = n and anv = pvPn , then we get |R,pn|k
summability (see [2]). Furthermore, if we take ϕn = n, anv =

pv
Pn

and pn = 1 for all values of n, then

ϕ−|A,pn|k summability reduces to |C, 1|k summability (see [4]).
Before stating the main theorem we must first introduce some further notations.
Given a normal matrix A = (anv), we associate two lower semimatrices Ā = (ānv) and  = (ânv) as
follows:

ānv =

n∑
i=v

ani, n,v = 0, 1, ...(7)

and

â00 = ā00 = a00, ânv = ānv − ān−1,v, n = 1, 2, ...(8)

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∑
v=0

anvsv =

n∑
v=0

ānvav(9)

and

∆̄An (s) =

n∑
v=0

ânvav.(10)

2. Known Result

Bor [3] has proved the following theorem for
∣∣N̄,pn∣∣k summability method.

Theorem 1. Let (pn) be a sequence of positive numbers such that

Pn = O(npn) as n →∞.(11)
If (Xn) is a positive monotonic non-decreasing sequence such that

|λm|Xm = O(1) as m →∞,(12)
m∑

n=1

nXn|∆2λn| = O(1) as m →∞(13)

and
m∑

n=1

pn
Pn
|tn|k = O(Xm) as m →∞,(14)

where

tn =
1

n + 1

n∑
v=1

vav,

then the series
∑
anλn is summable |N̄,pn|k, k ≥ 1.

3. Main Result

The aim of this paper is to generalize Theorem 1 to ϕ−|A,pn|k summability. Now we shall prove
the following theorem.
Theorem 2. Let A = (anv) be a positive normal matrix such that

ān0 = 1, n = 0, 1, ...,(15)

an−1,v ≥ anv, for n ≥ v + 1,(16)

ann = O

(
pn
Pn

)
,(17)

|ân,v+1| = O (v |∆vânv|) .(18)



68 ÖZARSLAN

Let (Xn) be a positive monotonic non-decreasing sequence and
(

ϕnpn
Pn

)
be a non-increasing sequence.

If conditions (12)-(13) of Theorem 1 and
m∑

n=1

ϕk−1n

(
pn
Pn

)k
|tn|k = O(Xm) as m →∞,(19)

are satisfied, then the series
∑
anλn is summable ϕ−|A,pn|k, k ≥ 1.

It should be noted that if we take ϕn =
Pn
pn

and anv =
pv
Pn

in Theorem 2, then we get Theorem 1.

In this case, condition (19) reduces to condition (14), condition (18) reduces to condition (11). Also,

the condition “
(

ϕnpn
Pn

)
is a non-increasing sequence” and the conditions (15)-(17) are automatically

satisfied. We require the following lemma for the proof of Theorem 2.
Lemma 1 ([3]). Under the conditions of Theorem 2, we have that

nXn|∆λn| = O(1) as n →∞,(20)
∞∑

n=1

Xn|∆λn| < ∞.(21)

4. Proof of Theorem 2

Let (In) denotes A-transform of the series
∑
anλn. Then, by (9) and (10), we have

∆̄In =

n∑
v=0

ânvavλv =

n∑
v=1

ânvλv
v

vav.

Applying Abel’s transformation to this sum, we get that

∆̄In =

n−1∑
v=1

∆v

(
ânvλv
v

) v∑
r=1

rar +
ânnλn
n

n∑
r=1

rar

=
n + 1

n
annλntn +

n−1∑
v=1

v + 1

v
∆v (ânv) λvtv

+

n−1∑
v=1

v + 1

v
ân,v+1∆λvtv +

n−1∑
v=1

ân,v+1λv+1
tv
v

= In,1 + In,2 + In,3 + In,4.

To complete the proof of Theorem 2, by Minkowski’s inequality, it is sufficient to show that
∞∑

n=1

ϕk−1n | In,r |
k< ∞, for r = 1, 2, 3, 4.(22)

First, by using Abel’s transformation, we have that
m∑

n=1

ϕk−1n |In,1|
k = O(1)

m∑
n=1

ϕk−1n a
k
nn|λn|

k|tn|k

= O(1)

m∑
n=1

ϕk−1n

(
pn
Pn

)k
|λn|k−1|λn||tn|k

= O(1)

m∑
n=1

ϕk−1n

(
pn
Pn

)k
|λn||tn|k

= O(1)

m−1∑
n=1

∆|λn|
n∑

v=1

ϕk−1v

(
pv
Pv

)k
|tv|k + O(1)|λm|

m∑
n=1

ϕk−1n

(
pn
Pn

)k
|tn|k

= O(1)

m−1∑
n=1

|∆λn|Xn + O(1)|λm|Xm

= O(1) as m →∞,



ON GENERALIZED ABSOLUTE MATRIX SUMMABILITY METHODS 69

by virtue of the hypotheses of Theorem 2 and Lemma 1. Now, applying Hölder’s inequality with
indices k and k′, where k > 1 and 1

k
+ 1

k′
= 1, as in In,1, we have that

m+1∑
n=2

ϕk−1n |In,2|
k = O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|∆v(ânv)| |λv| |tv|

)k

= O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|∆v(ânv)| |λv|
k |tv|

k

)
×

(
n−1∑
v=1

|∆v(ânv)|

)k−1

= O(1)

m+1∑
n=2

(
ϕnpn
Pn

)k−1 (n−1∑
v=1

|∆v(ânv)| |λv|
k |t

v
|k
)

= O(1)

m∑
v=1

|λv|k|tv|k
m+1∑

n=v+1

(
ϕnpn
Pn

)k−1
|∆v(ânv)|

= O(1)

m∑
v=1

|λv|k|tv|k
(
ϕvpv
Pv

)k−1 m+1∑
n=v+1

|∆v(ânv)|

= O(1)

m∑
v=1

|λv|k−1|λv||tv|kavv
(
ϕvpv
Pv

)k−1
= O(1)

m∑
v=1

ϕk−1v

(
pv
Pv

)k
|λv| |tv|

k

= O(1) as m →∞,

by virtue of the hypotheses of Theorem 2 and Lemma 1.
Now, using Hölder’s inequality we have that

m+1∑
n=2

ϕk−1n |In,3|
k = O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|ân,v+1||∆λv||tv|

)k

= O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

v|∆v(ânv)||∆λv||tv|

)k

= O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

(v|∆λv|)
k |tv|k|∆v(ânv)|

)
×

(
n−1∑
v=1

|∆v(ânv)|

)k−1

= O(1)

m+1∑
n=2

(
ϕnpn
Pn

)k−1 (n−1∑
v=1

(v|∆λv|)
k |tv|k|∆v(ânv)|

)

= O(1)

m∑
v=1

(v|∆λv|)
k |tv|k

m+1∑
n=v+1

(
ϕnpn
Pn

)k−1
|∆v(ânv)|

= O(1)

m∑
v=1

(v|∆λv|)
k−1

(v|∆λv|) |tv|k
(
ϕvpv
Pv

)k−1 m+1∑
n=v+1

|∆v(ânv)|

= O(1)

m∑
v=1

ϕk−1v

(
pv
Pv

)k
v|∆λv||tv|k

= O(1)

m−1∑
v=1

∆(v|∆λv|)
v∑

r=1

ϕk−1r

(
pr
Pr

)k
|tr|k + O(1)m|∆λm|

m∑
v=1

ϕk−1v

(
pv
Pv

)k
|tv|k

= O(1)

m−1∑
v=1

vXv|∆2λv| + O(1)
m−1∑
v=1

|∆λv|Xv + O(1)m|∆λm|Xm

= O(1) as m →∞,



70 ÖZARSLAN

by virtue of the hypotheses of Theorem 2 and Lemma 1.
Finally by using (18), as in In,1, we have that

m+1∑
n=2

ϕk−1n |In,4|
k ≤

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|ân,v+1||λv+1|
|tv|
v

)k

= O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|∆v(ânv)||λv+1||tv|

)k

= O(1)

m+1∑
n=2

ϕk−1n

(
n−1∑
v=1

|∆v(ânv)||λv+1|k|tv|k
)
×

(
n−1∑
v=1

|∆v(ânv)|

)k−1

= O(1)

m+1∑
n=2

(
ϕnpn
Pn

)k−1 (n−1∑
v=1

|∆v(ânv)||λv+1|k|tv|k
)

= O(1)

m∑
v=1

|λv+1|k−1|λv+1||tv|k
m+1∑

n=v+1

(
ϕnpn
Pn

)k−1
|∆v(ânv)|

= O(1)

m∑
v=1

|λv+1||tv|k
(
ϕvpv
Pv

)k−1 m+1∑
n=v+1

|∆v(ânv)|

= O(1)

m∑
v=1

|λv+1||tv|kavv
(
ϕvpv
Pv

)k−1
= O(1)

m∑
v=1

ϕk−1v

(
pv
Pv

)k
|λv+1| |tv|

k

= O(1) as m →∞,
by virtue of hypotheses of Theorem 2 and Lemma 1.
This completes the proof of Theorem 2.

5. Conclusions

It should be noted that, if we take ϕn =
Pn
pn

, then we get a theorem dealing with |A,pn|k summability.
Also, if we take anv =

pv
Pn

, then we have a result dealing with ϕ−|N̄,pn|k summability. Furthermore,
if we take anv =

pv
Pn

and pn = 1 for all values of n, then we get another result dealing with ϕ−|C, 1|k
summability. When we take ϕn = n, anv =

pv
Pn

and pn = 1 for all values of n, then we get a result

for |C, 1|k summability. Finally, if we take k = 1 and anv = pvPn , then we get a result for
∣∣N̄,pn∣∣

summability and in this case the condition “
(

ϕnpn
Pn

)
is a non-increasing sequence” is not needed.

References

[1] H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), 147-149.
[2] H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc. 113 (1991), 1009-

1012.
[3] H. Bor, On absolute summability factors, Proc. Amer. Math. Soc. 118 (1993), 71-75.

[4] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London

Math. Soc. 7 (1957), 113-141.
[5] G. H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.

[6] W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series. IV, Indian J.

Pure Appl. Math. 34 (11) (2003), 1547-1557.
[7] N. Tanovic̆-Miller, On strong summability, Glas. Mat. Ser. III 14 (34) (1979), 87-97.

Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey

∗Corresponding author: seyhan@erciyes.edu.tr