International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 2 (2017), 175-179
http://www.etamaths.com

FACTORS FOR ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY

OF INFINITE SERIES

HÜSEYİN BOR∗

Abstract. In this paper, we proved a general theorem dealing with absolute weighted arithmetic

mean summability factors of infinite series under weaker conditions. We have also obtained some

known results.

1. Introduction

Let
∑
an be a given infinite series with partial sums (sn). We denote by u

α
n the nth Cesàro mean

of order α, with α > −1, of the sequence (sn), that is (see [4])

uαn =
1

Aαn

n∑
v=0

Aα−1n−vsv, (1.1)

where

Aαn =
(α + 1)(α + 2)....(α + n)

n!
= O(nα), Aα−n = 0 for n > 0. (1.2)

A series
∑
an is said to be summable | C,α |k, k ≥ 1, if (see [5])

∞∑
n=1

nk−1 | uαn −u
α
n−1 |

k< ∞. (1.3)

If we take α=1, then we obtain | C, 1 |k summability. Let (pn) be a sequence of positive numbers
such that Pn =

∑n
v=0 pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1). The sequence-to-sequence

transformation

wn =
1

Pn

n∑
v=0

pvsv (1.4)

defines the sequence (wn) of the weighted arithmetic mean or simply the (N̄,pn) mean of the sequence
(sn), generated by the sequence of coefficients (pn) (see [6]). The series

∑
an is said to be summable

| N̄,pn |k, k ≥ 1, if (see [1])
∞∑
n=1

(Pn/pn)
k−1 | wn −wn−1 |k< ∞. (1.5)

If we take pn = 1 for all values of n, then we obtain | C, 1 |k summability. Also if we take k = 1, then
we obtain | N̄,pn | summability (see [11]). For any sequence (λn) we write that ∆λn = λn −λn+1.

2. Known Result

The following theorem is known dealing with | N̄,pn |k summability factors of infinite series.

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99.
Key words and phrases. weighted arithmetic mean; absolute summability; summability factors; infinite series; Hölder

inequality; Minkowski inequality.

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.

175



176 BOR

Theorem 2.1. [2] Let (Xn) be a positive non-decreasing sequence and suppose that there exists
sequences (βn) and (λn) such that

| ∆λn |≤ βn, (2.1)
βn → 0 as n →∞, (2.2)
∞∑
n=1

n | ∆βn | Xn < ∞, (2.3)

| λn | Xn = O(1). (2.4)
If

m∑
n=1

| sn |k

n
= O(Xm) as m →∞, (2.5)

and (pn) is a sequence such that

Pn = O(npn), (2.6)

Pn∆pn = O(pnpn+1), (2.7)

then the series
∑∞
n=1 an

Pnλn
npn

is summable | N̄,pn |k, k ≥ 1.

Remark 2.1. It should be noted that, under the conditions on the sequence (λn) we have that (λn) is
bounded and ∆λn = O(1/n) [2].

3. Main Result

The aim of this paper is to prove Theorem 2.1 under weaker conditions. Now, we shall prove the
following theorem.

Theorem 3.1. Let (Xn) be a positive non-decreasing sequence. If the sequences (Xn), (βn), (λn), and
(pn) satisfy the conditions (2.1)-(2.4), (2.6)-(2.7), and

m∑
n=1

| sn |k

nXk−1n
= O(Xm) as m →∞, (3.1)

then the series
∑∞
n=1 an

Pnλn
npn

is summable | N̄,pn |k, k ≥ 1.

Remark 3.1. It should be noted that condition (3.1) is the same as condition (2.5) when k=1. When
k > 1, condition (3.1) is weaker than condition (2.5) but the converse is not true. As in [10], we can
show that if (2.5) is satisfied, then we get

m∑
n=1

| sn |k

nXk−1n
= O(

1

Xk−11
)

m∑
n=1

| sn |k

n
= O(Xm) as m →∞.

To show that the converse is false when k > 1, as in [3], the following example is sufficient. We can
take Xn = n

δ, 0 < δ < 1, and then construct a sequence (un) such that

un =
|sn|k

nXn
k−1 = Xn −Xn−1,

hence
m∑
n=1

|sn|k

nXn
k−1 = Xm = m

δ,

and so
m∑
n=1

|sn|k

n
=

m∑
n=1

(Xn −Xn−1)Xk−1n =
m∑
n=1

(nδ − (n− 1)δ)nδ(k−1)

≥ δ
m∑
n=1

nδ−1nδ(k−1) = δ

m∑
n=1

nδk−1 ∼
mδk

k
as m →∞.



ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY 177

It follows that

1

Xm

m∑
n=1

|sn|k

n
→∞ as m →∞

provided k > 1. This shows that (2.5) implies (3.1) but not conversely.
We require the following lemmas for the proof of Theorem 3.1.

Lemma 3.1. [7] Under the conditions on (Xn), (βn) and (λn) as as expressed in the statement of
the theorem, we have the following;

nXnβn = O(1), (3.2)

∞∑
n=1

βnXn < ∞. (3.3)

Lemma 3.2. [9] If the conditions (2.6) and (2.7) are satisfied, then ∆
(
Pn
npn

)
= O

(
1
n

)
.

4. Proof of Theorem 3.1

Proof. Let (Tn) be the sequence of (N̄,pn) mean of the series
∑∞
n=1

anPnλn
npn

. Then, by definition, we

have

Tn =
1

Pn

n∑
v=1

pv

v∑
r=1

arPrλr
rpr

=
1

Pn

n∑
v=1

(Pn −Pv−1)
avPvλv
vpv

.

Then we get that

Tn −Tn−1 =
pn

PnPn−1

n∑
v=1

Pv−1Pvavλv
vpv

, n ≥ 1, (P−1 = 0).

By using Abel’s transformation, we have that

Tn −Tn−1 =
pn

PnPn−1

n−1∑
v=1

sv∆

(
Pv−1Pvλv

vpv

)
+
λnsn
n

=
snλn
n

+
pn

PnPn−1

n−1∑
v=1

sv
Pv+1Pv∆λv
(v + 1)pv+1

+
pn

PnPn−1

n−1∑
v=1

Pvsvλv∆

(
Pv
vpv

)
−

pn
PnPn−1

n−1∑
v=1

svPvλv
1

v

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

To complete the proof of the Theorem 3.1, by Minkowski’s inequality, it is sufficient to show that

∞∑
n=1

(
Pn
pn

)k−1
| Tn,r |k< ∞, for r = 1, 2, 3, 4. (4.1)

Applying Abel’s transformation, we have that

m∑
n=1

(
Pn
pn

)k−1
| Tn,1 |k=

m∑
n=1

(
Pn
npn

)k−1
| λn |k−1| λn |

| sn |k

n
= O(1)

m∑
n=1

| sn |k

n

(
1

Xn

)k−1
| λn |

= O(1)

m−1∑
n=1

∆ | λn |
n∑
v=1

| sv |k

vXv
k−1 + O(1) | λm |

m∑
n=1

| sn |k

nXn
k−1 = O(1)

m−1∑
n=1

| ∆λn | Xn + O(1) | λm | Xm

= O(1)

m−1∑
n=1

βnXn + O(1) | λm | Xm = O(1), as m →∞,



178 BOR

by the hypotheses of Theorem 3.1 and Lemma 3.1. Now, by using (2.6) and applying Hölder’s inequal-
ity, we obtain that

m+1∑
n=2

(
Pn
pn

)k−1
| Tn,2 |k= O(1)

m+1∑
n=2

pn
PnP

k
n−1
|
n−1∑
v=1

Pvsv∆λv |k= O(1)
m+1∑
n=2

pn
PnP

k
n−1

{
n−1∑
v=1

Pv
pv
| sv | pv | ∆λv |

}k

= O(1)

m+1∑
n=2

pn
PnPn−1

n−1∑
v=1

(
Pv
pv

)k
| sv |k pvβvk ×

(
1

Pn−1

n−1∑
v=1

pv

)k−1

= O(1)

m∑
v=1

(
Pv
pv

)k
| sv |k pvβvk

m+1∑
n=v+1

pn
PnPn−1

= O(1)

m∑
v=1

(
Pv
pv

)k−1
βv
k−1βv | sv |k = O(1)

m∑
v=1

(vβv)
k−1βv | sv |k

= O(1)

m∑
v=1

(
1

Xv

)k−1
βv | sv |k= O(1)

m∑
v=1

vβv
| sv |k

vXv
k−1

= O(1)

m−1∑
v=1

∆(vβv)

v∑
r=1

| sr |k

rXr
k−1 + O(1)mβm

m∑
v=1

| sv |k

vXv
k−1 = O(1)

m−1∑
v=1

| ∆(vβv) | Xv + O(1)mβmXm

= O(1)

m−1∑
v=1

| (v + 1)∆βv −βv | Xv + O(1)mβmXm

= O(1)

m−1∑
v=1

v | ∆βv | Xv + O(1)
m−1∑
v=1

Xvβv + O(1)mβmXm = O(1),

as m →∞, by the hypotheses of the Theorem 3.1 and Lemma 3.1. Again, as in Tn,1, we have that
m+1∑
n=2

(
Pn
pn

)k−1
| Tn,3 |k=

m+1∑
n=2

(
Pn
pn

)k−1
|

pn
PnPn−1

n−1∑
v=1

Pvsvλv∆

(
Pv
vpv

)
|k

= O(1)

m+1∑
n=2

pn
PnP

k
n−1

{
n−1∑
v=1

Pv | sv || λv |
1

v

}k
= O(1)

m+1∑
n=2

pn
PnP

k
n−1

{
n−1∑
v=1

(
Pv
pv

)
pv | sv || λv |

1

v

}k

= O(1)

m+1∑
n=2

pn
PnPn−1

n−1∑
v=1

(
Pv
vpv

)k
pv | sv |k| λv |k ×

{
1

Pn−1

n−1∑
v=1

pv

}k−1

= O(1)

m∑
v=1

(
Pv
vpv

)k
| sv |k pv | λv |k

m+1∑
n=v+1

pn
PnPn−1

= O(1)

m∑
v=1

(
Pv
vpv

)k
pv | sv |k| λv |k

1

Pv
.
v

v

= O(1)

m∑
v=1

(
Pv
vpv

)k−1
| λv |k−1| λv |

| sv |k

v
= O(1)

m∑
v=1

(
1

Xv

)k−1
| λv |

| sv |k

v
= O(1)

m∑
v=1

| λv |
| sv |k

vXv
k−1

= O(1)

m−1∑
v=1

Xvβv + O(1)Xm | λm |= O(1), as m →∞,

by the hypotheses of the Theorem 3.1, Lemma 3.1 and Lemma 3.2. Finally, using Hölder’s inequality,
as in Tn,3, we have get

m+1∑
n=2

(
Pn
pn

)k−1
| Tn,4 |k=

m+1∑
n=2

pn
PnP

k
n−1
|
n−1∑
v=1

sv
Pv
v
λv |k

=

m+1∑
n=2

pn
PnP

k
n−1
|
n−1∑
v=1

sv
Pv
vpv

pvλv |k≤
m+1∑
n=2

pn
PnPn−1

n−1∑
v=1

| sv |k
(
Pv
vpv

)k
pv | λv |k ×

(
1

Pn−1

n−1∑
v=1

pv

)k−1



ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY 179

= O(1)

m∑
v=1

(
Pv
vpv

)k
| sv |k pv | λv |k

1

Pv
.
v

v
= O(1)

m∑
v=1

(
Pv
vpv

)k−1
| λv |k−1| λv |

| sv |k

v

= O(1)

m∑
v=1

(
1

Xv

)k−1
| λv |

| sv |k

v
= O(1)

m∑
v=1

| λv |
| sv |k

vXv
k−1

= O(1)

m−1∑
v=1

Xvβv + O(1)Xm | λm |= O(1), as m →∞.

This completes the proof of Theorem 3.1. �

5. Conclusions

It should be noted that if we take pn = 1 for all n, then we obtain a known result of Mishra and
Srivastava dealing with | C, 1 |k summability factors of infinite series (see [8]). Also, if we set k = 1,
then we have a known result of Mishra and Srivastava concerning the | N̄,pn | summability factors of
infinite series (see [9]).

References

[1] H. Bor, On two summability methods, Math. Proc. Camb. Philos Soc. 97 (1985), 147-149.

[2] H. Bor, A note on | N̄, pn |k summability factors of infinite series, Indian J. Pure Appl. Math. 18 (1987), 330-336.
[3] H. Bor, Quasi-monotone and almost increasing sequences and their new applications, Abstr. Appl. Anal. 2012, Art.

ID 793548, 6 pp.

[4] E. Cesàro, Sur la multiplication des séries, Bull. Sci. Math. 14 (1890), 114-120.
[5] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London

Math. Soc., 7 (1957), 113-141.

[6] G. H. Hardy, Divergent Series, Clarendon Press, Oxford, (1949).
[7] K. N. Mishra, On the absolute Nörlund summability factors of infinite series, Indian J. Pure Appl. Math. 14 (1983),

40-43.

[8] K. N. Mishra and R. S. L. Srivastava, On the absolute Cesàro summability factors of infinite series, Portugal. Math.
42 (1983/84), 53-61.

[9] K. N. Mishra and R. S. L. Srivastava, On | N̄, pn | summability factors of infinite series, Indian J. Pure Appl. Math.
15 (1984), 651-656.

[10] W. T. Sulaiman, A note on |A|k summability factors of infinite series, Appl. Math. Comput. 216 (2010), 2645-2648.
[11] G. Sunouchi, Notes on Fourier analysis. XVIII. Absolute summability of series with constant terms, Tôhoku Math.

J. (2), 1 (1949), 57-65.

P. O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey

∗Corresponding author: hbor33@gmail.com


	1. Introduction
	2. Known Result
	3. Main Result
	4. Proof of Theorem 3.1
	5. Conclusions
	References