International Journal of Analysis and Applications
ISSN 2291-8639
Volume 15, Number 1 (2017), 108-113
http://www.etamaths.com

A NOTE ON ABSOLUTE CESÀRO ϕ−|C, 1; δ; l|k SUMMABILITY FACTOR

SMITA SONKER1, XH. Z. KRASNIQI2,∗ AND ALKA MUNJAL1

Abstract. A positive non-decreasing sequence has been used to establish a theorem on a minimal

set of sufficient conditions for an infinite series to be absolute Cesàro ϕ−|C, 1; δ; l|k summable. For
some well-known applications, suitable conditions have been applied on the presented theorem for

obtaining the sub-result of the presented theorem.

1. Introduction

Let {sn} be a sequence of partial sums of the series
∞∑
n=0

an and n
th mean of {sn} is given by tn s.t.

tn =

∞∑
k=0

tnksk (1.1)

where {tnk} is the sequence of the coefficients of the matrix. If sequence of the means {tn} satisfied
the following conditions:

lim
n→∞

tn = s, (1.2)

and
∞∑
n=1

| tn − tn−1| < ∞, (1.3)

then the series
∞∑
n=0

an is said to be absolute summable. If τn represent the n
th (C, 1) means of the

sequence (nan), then series
∞∑
n=0

an is said to be summable |C, 1|k,k ≥ 1 [9], if

∞∑
n=1

1

n
|τn|k < ∞. (1.4)

If the sequence {τn} satisfied the condition:
∞∑
n=1

ϕk−1n
nk
|τn|k < ∞, (1.5)

then the series
∞∑
n=0

an is said to be summable ϕ−|C, 1|k, k ≥ 1, and if the sequence {τn} satisfied the

following condition:
∞∑
n=1

ϕk−1n
nk−δk

|τn|k < ∞, (1.6)

then the series
∞∑
n=0

an is ϕ−|C, 1; δ|k, summable, where k ≥ 1, δ ≥ 0 and (ϕn) be a sequence of positive

real numbers.

Received 27th May, 2017; accepted 1st August, 2017; published 1st September, 2017.
2010 Mathematics Subject Classification. 40F05, 40D15, 40G05.

Key words and phrases. absolute summability; infinite series; ϕ−|C, 1; δ; l|k summability; sequence space.
c©2017 Authors retain the copyrights of

their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

108



ABSOLUTE CESÀRO ϕ−|C, 1; δ; l|k SUMMABILITY 109

For ϕ−|C, 1; δ; l|k summability, the infinite series
∞∑
n=0

an satisfied

∞∑
n=1

ϕ
l(k−1)
n

nl(k−δk)
|τn|k < ∞ (1.7)

where k ≥ 1, δ ≥ 0 and l is a real number.
Note: If we take l = 1, then ϕ − |C, 1; δ; l|k reduces to ϕ − |C, 1,δ|k, if ϕ = n, then ϕ − |C, 1; δ|k
summability reduces to |C, 1; δ|k summability and if δ = 0, then |C, 1; δ|k reduces to |C, 1|k.

In 1972, Mazhar [8] determined the minimal set of sufficient conditions for an infinite series to be
absolute |C, 1|k summable. This result became an essence of many results found in previous years.
In 1980, Balci [10] defined Absolute ϕ-summability factors and determined a very interesting result.
Bor gave a number of theorems on absolute summability by a generalization of Mazhar [8] results. In
1993 [2] and 1994 [3], he used |N̄,pn|k summability for enhancement of the results of Mazhar [8] and
apply it on Fourier series. Özarslan [4] generalized the result of Bor [1] by a more general absolute
ϕ−|C,α|k summability and in [6], he used absolute matrix ϕ−|A,δ|k summability and improve some
well-known results. Concerning the ϕ−|N,pn|k summability factors, Saxena [7] gave a general theorem
for an infinite series.

2. Known results

Absolute ϕ−|C, 1|k summability has been used by Özarslan [5] to establish the following theorem.

Theorem 2.1. Let ϕn be a sequence of positive real numbers, if

λm = O(1), m →∞, (2.1)
m∑
n=1

n log n|∆2λn| = O(1), (2.2)

m∑
v=1

ϕk−1v
vk
|tv|k = O(log m), as m →∞, (2.3)

m∑
n=v

ϕk−1n
nk+1

= O

(
ϕk−1v
vk

)
. (2.4)

Then the infinite series
∑
anλn is ϕ−|C, 1|k summable for k ≥ 1.

3. Main results

Generalized Cesáro ϕ−|C, 1; δ; l|k summability and a positive non-decreasing sequence have been
used to moderate the conditions of Özarslan [5] results for an infinite series.

Theorem 3.1. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing
sequence satisfying the following conditions:

λm = O(1), m →∞, (3.1)
m∑
n=1

n log n|∆2λn| = O(1), (3.2)

m∑
v=1

ϕ
l(k−1)
v

vl(k−δk)
|tv|k = O(log m.µm) as m →∞, (3.3)

m∑
n=v

ϕ
l(k−1)
n

n1+l(k−δk)
= O

(
ϕ
l(k−1)
v

vl(k−δk)

)
, (3.4)

n log nµn∆
( 1
µn

)
= O(1). (3.5)

Then the infinite series
∑
anλn/µn is ϕ−|C, 1; δ; l|k summable for k ≥ 1, δ ≥ 0 and l is a real number.



110 SONKER, KRASNIQI AND MUNJAL

4. Proof of the Theorem

Let Tn be the n
th (C, 1) mean of the sequence (nanλn/µn). The series is ϕ−|C, 1; δ; l|k summable,

if

∞∑
n=1

ϕ
l(k−1)
n

nl(k−δk)
|Tn|k < ∞. (4.1)

Applying Able’s transformation, we have

Tn =
1

n + 1

n∑
v=1

vavλv
µv

=
1

n + 1

(
n−1∑
v=1

(
v∑
r=1

rar

)
∆

(
λv
µv

)
+

(
λv
µv

)
n∑
v=1

vav

)

=
1

n + 1

n−1∑
v=1

(v + 1)tv

( 1
µv

)
∆λv +

1

n + 1

n−1∑
v=1

(v + 1)tvλv+1∆

(
1

µv

)
+
tnλn
µn

= Tn,1 + Tn,2 + Tn,3. (4.2)

Using Minkowski’s inequality,

|Tn|k = |Tn,1 + Tn,2 + Tn,3|k < 3k
(
|Tn,1|k + |Tn,2|k + |Tn,3|k

)
. (4.3)

In order to complete the proof of the theorem, it is sufficient to show that

∞∑
n=1

ϕ
l(k−1)
n

nl(k−δk)
|Tn,r|k < ∞, for r = 1, 2, 3. (4.4)

By using Hölder’s inequality and Abel’s transformation, we have

m∑
n=2

ϕ
l(k−1)
n

nl(k−δk)
|Tn,1|k =

m∑
n=2

ϕ
l(k−1)
n

nl(k−δk)

∣∣∣∣∣ 1n + 1
n−1∑
v=1

(v + 1)tv
∆λv
µv

∣∣∣∣∣
k

= O(1)

m∑
n=2

ϕ
l(k−1)
n

nk+l(k−δk)

(
n−1∑
v=1

v|tv|
|∆λv|
µv

)k

= O(1)

m∑
n=2

ϕ
l(k−1)
n

nk+l(k−δk)

n−1∑
v=1

v
|∆λv|
µv
|tv|k

(
n−1∑
v=1

v
|∆λv|
µv

)k−1

= O(1)

m∑
n=2

ϕ
l(k−1)
n

n1+l(k−δk)

(
n−1∑
v=1

v
|∆λv|
µv
|tv|k

)

= O(1)

m∑
v=1

v
|∆λv|
µv
|tv|k

(
m∑
n=v

ϕ
l(k−1)
n

n1+l(k−δk)

)

= O(1)

m∑
v=1

v
|∆λv|
µv
|tv|k

ϕ
l(k−1)
v

vl(k−δk)



ABSOLUTE CESÀRO ϕ−|C, 1; δ; l|k SUMMABILITY 111

= O(1)

m−1∑
v=1

∣∣∣∣∣∆
(
v
|∆λv|
µv

)∣∣∣∣∣
v∑
r=1

ϕ
l(k−1)
r

rl(k−δk)
|tr|k + m

|∆λm|
µm

v∑
r=1

ϕ
l(k−1)
r

rl(k−δk)
|tr|k

= O(1)

m−1∑
v=1

|∆λv|
µv

log v.µv +

m−1∑
v=1

(v + 1)∆

(
|∆λv|
µv

)
log v.µv

+ m
|∆λm|
µm

log m.µm

= O(1)

m−1∑
v=1

|∆λv| log v +
m−1∑
v=1

(v + 1)
1

µv
|∆2λv| log v.µv

+

m−1∑
v=1

(v + 1)|∆λv+1|∆
(

1

µv

)
log v.µv + m|∆λm| log m

= O(1). (4.5)

m∑
n=2

ϕ
l(k−1)
n

nl(k−δk)
|Tn,2|k =

m∑
n=2

ϕ
l(k−1)
n

nl(k−δk)

∣∣∣∣∣ 1n + 1
n−1∑
v=1

λv+1(v + 1)tv∆

(
1

µv

)∣∣∣∣∣
k

= O(1)

m∑
n=2

ϕ
l(k−1)
n

nk+l(k−δk)

(
n−1∑
v=1

vλv+1|tv|∆
(

1

µv

))k

= O(1)

m∑
n=2

ϕ
l(k−1)
n

nk+l(k−δk)

n−1∑
v=1

vλv+1∆

(
1

µv

)
|tv|k

(
n−1∑
v=1

vλv+1∆

(
1

µv

))k−1

= O(1)

m∑
n=2

ϕ
l(k−1)
n

n1+l(k−δk)

(
n−1∑
v=1

vλv+1∆

(
1

µv

)
|tv|k

)

= O(1)

m−1∑
v=1

∣∣∣∣∣vλv+1∆
(

1

µv

)∣∣∣∣∣
v∑
r=1

ϕ
l(k−1)
r

r1+l(k−δk)
|tr|k

+ mλm+1∆

(
1

µm

) m∑
r=1

ϕ
l(k−1)
r

r1+l(k−δk)
|tr|k

= O(1)

m−1∑
v=1

λv+1∆

(
1

µv

)
µv log v +

m−1∑
v=1

(v + 1)∆

(
λv+1∆

(
1

µv

))
µv log v

+ mλm+1∆

(
1

µm

)
log m.µm

= O(1)

m−1∑
v=1

λv+1 log v +

m−1∑
v=1

(v + 1)∆λv+1∆

(
1

µv

)
µv log v

+

m−1∑
v=1

(v + 1)λv+2∆
2

(
1

µv

)
µv log v + mλm+1 log m

= O(1) as m →∞. (4.6)

m∑
n=1

ϕ
l(k−1)
n

nl(k−δk)
|Tn,3|k =

m∑
n=1

ϕ
l(k−1)
n

nl(k−δk)

∣∣∣∣∣tnλnµn
∣∣∣∣∣
k

= O(1)

m∑
n=1

ϕ
l(k−1)
n

nl(k−δk)
|tn|k

∣∣∣∣∣
∞∑
v=n

∆

(
λv
µv

)∣∣∣∣∣



112 SONKER, KRASNIQI AND MUNJAL

= O(1)

∞∑
v=1

∣∣∣∣∣∆
(
λv
µv

)∣∣∣∣∣
v∑

n=1

ϕ
l(k−1)
n

nl(k−δk)
|tn|k

= O(1)

∞∑
v=1

1

µv
∆λv log v.µv +

∞∑
v=1

λv+1∆

(
1

µv

)
log v.µv

= O(1). (4.7)

Collecting (4.2) - (4.7), we have
∞∑
n=1

ϕk−1n
nk−δk

|Tn|k < ∞. (4.8)

Hence proof of the theorem is completed.

5. Corollaries

Corollary 5.1. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing
sequence satisfying (3.1)-(3.2), (3.5) and following conditions:

m∑
v=1

ϕk−1v
vk−δk

|tv|k = O(log m.µm) as m →∞, (5.1)

m∑
n=v

ϕk−1n
n1+k−δk

= O

(
ϕk−1v
vk−δk

)
. (5.2)

Then the infinite series
∑
anλn/µn is ϕ−|C, 1; δ|k summable for k ≥ 1 and δ ≥ 0.

Proof. By using specific value l = 1 in Theorem 3.1, we will get (5.1) and (5.2). We omit the details
as the proof is similar to that of Theorem 3.1 and we use (5.1) and (5.2) instead of (3.3) and (3.4). �

Corollary 5.2. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing
sequence satisfying (3.1)-(3.2), (3.5) and following conditions:

m∑
v=1

ϕk−1v
vk
|tv|k = O(log m.µm) as m →∞, (5.3)

m∑
n=v

ϕk−1n
n1+k

= O

(
ϕk−1v
vk

)
. (5.4)

Then the infinite series
∑
anλn/µn is ϕ−|C, 1; δ|k summable for k ≥ 1 and δ ≥ 0.

Proof. By using specific value l = 1 and δ = 0 in Theorem 3.1, we will get (5.3) and (5.4). We omit
the details as the proof is similar to that of Theorem 3.1 and we use (5.3) and (5.4) instead of (3.3)
and (3.4). �

Hence theorem 3.1 is a generalization of above Corollaries.

6. Conclusion

The aim of this research article is to formulate the problem of generalization of absolute Cesáro
(ϕ −|C, 1; δ; l|k, k ≥ 1, δ ≥ 0 and l is a real number) summability factor of infinite series which is a
motivation for the researchers, interested in theoretical studies of an infinite series. Further, this study
has a number of direct applications in rectification of signals in FIR filter (Finite impulse response
filter) and IIR filter (Infinite impulse response filter). In a nut shell, the absolute summability methods
have vast potential in dealing with the problems based on infinite series.

Acknowledgments. The authors express their sincere gratitude to the Department of Science and
Technology (India) for providing financial support to the second author under INSPIRE Scheme (In-
novation in Science Pursuit for Inspired Research Scheme).



ABSOLUTE CESÀRO ϕ−|C, 1; δ; l|k SUMMABILITY 113

References

[1] H. Bor, Absolute summability factors, Atti Sem. Mat. Fis. Univ. Modena., 39 (1991), 419-422.

[2] H. Bor, On absolute summability factors, Proc. Amer. Math. Soc., 118(1) (1993), 71-75.
[3] H. Bor, On the absolute Riesz summability factors, Rocky Mountain J. Math., 24(4) (1994), 1263-1271.

[4] H. S. Özarslan, A note on absolute summability factors, Proc. Indian Acad. Sci., 113 (2003), 165-169.

[5] H. S. Özarslan, On absolute Cesáro summability factors of infinite series, Com. Math. Anal., 3(1) (2007), 53-56.

[6] H. S. Özarslan and T. Ari, Absolute matrix summability methods, Applied Math. Lett., 24 (2011), 2102-2106.
[7] S. K. Saxena, On Nörlund summability factors of infinite series, Int. J. Math. Analysis, 22(2) (2008), 1097 - 1102.

[8] S. M. Mazhar, On (C, 1) summability factors of infinite series, Indian J. Math., 14 (1972), 45-48.

[9] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London
Math. Sci., 7 (1957), 113-141.

[10] M. Balci, Absolute ϕ-summability factors, Comm. Fac. Sci. Univ. Ankara, Ser. A1, 29 (1980), 63-80.

1Department of Mathematics, National Institute of Technology, Kurukshetra-136119, Haryana, India

2Faculty of Education, University of Prishtina ”Hasan Prishtina”, Avenue ”Mother Theresa” 5, 10000

Prishtina, Kosovo

∗Corresponding author: xhevat.krasniqi@uni-pr.edu


	1. Introduction
	2. Known results
	3. Main results
	4. Proof of the Theorem
	5. Corollaries
	6. Conclusion
	References