International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 2 (2017), 134-139
http://www.etamaths.com

AN APPLICATION OF δ-QUASI MONOTONE SEQUENCE

HİKMET SEYHAN ÖZARSLAN∗

Abstract. In this paper, a known theorem dealing with |A,pn|k summability method of infinite
series has been generalized to |A,pn; δ|k summability method. Also, some results have been obtained.

1. Introduction

A sequence (dn) is said to be δ-quasi-monotone, if dn → 0, dn > 0 ultimately and ∆dn ≥ −δn,
where ∆dn=dn −dn+1 and δ = (δn) is a sequence of positive numbers (see [1]). Let

∑
an be a given

infinite series with 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) . (1.1)

The sequence-to-sequence transformation

zn =
1

Pn

n∑
v=0

pvsv (1.2)

defines the sequence (zn) of the Riesz mean or simply 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 [2])

∞∑
n=1

(
Pn
pn

)k−1
|∆zn−1|k < ∞, (1.3)

where

∆zn−1 = −
pn

PnPn−1

n∑
v=1

Pv−1av, n ≥ 1.

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, ... (1.4)

The series
∑
an is said to be summable |A,pn; δ|k, k ≥ 1 and δ ≥ 0, if (see [6])

∞∑
n=1

(
Pn
pn

)δk+k−1
|∆̄An(s)|k < ∞, (1.5)

where

∆̄An(s) = An(s) −An−1(s).
If we set δ = 0, then |A,pn; δ|k summability reduces to |A,pn|k summability (see [8]). If we take
anv =

pv
Pn

and δ = 0, then |A,pn; δ|k summability reduces to |N̄,pn|k summability.

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99.
Key words and phrases. summability factors; absolute matrix summability; quasi-monotone sequences; 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.

134



AN APPLICATION OF δ-QUASI MONOTONE SEQUENCE 135

In the special case δ = 0 and pn = 1 for all n, |A,pn; δ|k summability is the same as |A|k summability
(see [9]). Also, if we take anv =

pv
Pn

, then |A,pn; δ|k summability is the same as |N̄,pn; δ|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, ... (1.6)

and

â00 = ā00 = a00, ânv = ānv − ān−1,v, n = 1, 2, ... (1.7)

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 (1.8)

and

∆̄An (s) =

n∑
v=0

ânvav. (1.9)

2. Known Results

In [3], Bor has proved the following theorem dealing with |N̄,pn|k summability.

Theorem 2.1. Let (Xn) be a positive non-decreasing sequence, (λn) → 0 as n → ∞ and (pn) be a
sequence of positive numbers such that

Pn = O(npn) as n →∞. (2.1)
Suppose that there exist a sequence of numbers (Bn) which is δ-quasi monotone with∑
nXnδn < ∞,

∑
BnXn is convergent and |∆λn| ≤ |Bn| for all n. If

m∑
n=1

pn
Pn
|tn|k = O(Xm) as m →∞, (2.2)

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

Later on, in [7], Özarslan and Şakar have proved the following theorem dealing with |A,pn|k summa-
bility factors of infinite series.

Theorem 2.2. Let A = (anv) be a positive normal matrix such that

ān0 = 1, n = 0, 1, ..., (2.3)

an−1,v ≥ anv for n ≥ v + 1, (2.4)

ann = O

(
pn
Pn

)
, (2.5)

|ân,v+1| = O (v |∆vânv|) . (2.6)
If (Xn) is a positive non-decreasing sequence and the conditions of Theorem 2.1 are satisfied, then the
series

∑
anλn is summable |A,pn|k, k ≥ 1.



136 ÖZARSLAN

3. Main Result

The purpose of this paper is to generalize Theorem 2.2 for |A,pn; δ|k summability.
Now, we shall prove the following more general theorem.

Theorem 3.1. Let A = (anv) be a positive normal matrix such that

m+1∑
n=v+1

(
Pn
pn

)δk
|∆vânv| = O

{(
Pv
pv

)δk−1}
as m →∞. (3.1)

If all conditions of Theorem 2.2 with condition (2.2) replaced by:

m∑
n=1

(
Pn
pn

)δk−1
|tn|k = O(Xm) as m →∞, (3.2)

are satisfied, then the series
∑
anλn is summable |A,pn; δ|k , k ≥ 1 and 0 ≤ δ < 1/k.

We require the following lemmas for the proof of Theorem 3.1.

Lemma 3.1. ( [3]). Under the conditions of Theorem 3.1, we have that

|λn|Xn = O (1) as n →∞. (3.3)

Lemma 3.2. ( [3]). Let (Xn) be a positive non-decreasing sequence. If (Bn) is δ-quasi monotone with∑
nXnδn < ∞ and

∑
BnXn is convergent, then

nBnXn = O (1) as n →∞, (3.4)
∞∑
n=1

nXn|∆Bn| < ∞. (3.5)

4. Proof of Theorem 3.1

Let (In) denotes A-transform of the series
∑
anλn. Then, by (1.8) and (1.9), 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∑
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

+
n + 1

n
annλntn

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

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

∞∑
n=1

(
Pn
pn

)δk+k−1
|In,r|

k
< ∞, for r = 1, 2, 3, 4.



AN APPLICATION OF δ-QUASI MONOTONE SEQUENCE 137

First, when k > 1, applying Hölder’s inequality with indices k and k
′
, where 1

k
+ 1

k
′ = 1, we have that

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,1|k = O(1)

m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

|∆v(ânv)||λv||tv|

)k

= O(1)

m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

|∆v(ânv)||λv|k|tv|k
)

×

(
n−1∑
v=1

|∆v(ânv)|

)k−1
.

By (1.6) and (1.7), we have that

∆v(ânv) = ânv − ân,v+1 = ānv − ān−1,v − ān,v+1 + ān−1,v+1 = anv −an−1,v.

Thus using (1.6), (2.3) and (2.4)

n−1∑
v=1

|∆v(ânv)| =
n−1∑
v=1

(an−1,v −anv) ≤ ann.

Hence, we get

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,1|k = O(1)

m+1∑
n=2

(
Pn
pn

)δk (n−1∑
v=1

|∆v(ânv)||λv|k|tv|k
)

= O(1)

m∑
v=1

|λv|k−1|λv||tv|k
m+1∑
n=v+1

(
Pn
pn

)δk
|∆v(ânv)|

= O(1)

m∑
v=1

(
Pv
pv

)δk−1
|λv||tv|k

= O(1)

m−1∑
v=1

∆|λv|
v∑
r=1

(
Pr
pr

)δk−1
|tr|k

+ O(1)|λm|
m∑
v=1

(
Pv
pv

)δk−1
|tv|k

= O(1)

m−1∑
v=1

|∆λv|Xv + O(1)|λm|Xm

= O(1)

m−1∑
v=1

BvXv + O(1)|λm|Xm

= O(1) as m →∞,

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.1.



138 ÖZARSLAN

Again, by using Hölder’s inequality, we have that

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,2|

k
= O(1)

m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

|ân,v+1||∆λv||tv|

)k

= O(1)

m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

v |∆v(ânv)| |Bv||tv|k
)

×

(
n−1∑
v=1

v |∆v(ânv)| |Bv|

)k−1
.

By using (3.4), we get

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,2|

k
= O(1)

m+1∑
n=2

(
Pn
pn

)δk (n−1∑
v=1

v |∆v(ânv)| |Bv||tv|k
)

= O(1)

m∑
v=1

v|Bv||tv|k
m+1∑
n=v+1

(
Pn
pn

)δk
|∆v(ânv)|

= O(1)

m∑
v=1

(
Pv
pv

)δk−1
v|Bv||tv|k.

Now, applying Abel’s transformation to this sum, we have that

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,2|

k
= O(1)

m−1∑
v=1

|∆ (v|Bv|)|
v∑
r=1

(
Pr
pr

)δk−1
|tr|k

+ O(1)m |Bm|
m∑
v=1

(
Pv
pv

)δk−1
|tv|k

= O(1)

m−1∑
v=1

v|∆Bv|Xv + O(1)
m−1∑
v=1

BvXv + O(1)mBmXm

= O(1) as m →∞,

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2.
Also, as in In,1, we have that

m+1∑
n=2

(
Pn
pn

)δk+k−1
|In,3|

k ≤
m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

|ân,v+1||λv+1|
|tv|
v

)k

= O(1)

m+1∑
n=2

(
Pn
pn

)δk+k−1 (n−1∑
v=1

|∆v(ânv)| |λv+1|k|tv|k
)

×

(
n−1∑
v=1

|∆v(ânv)|

)k−1

= O(1)

m∑
v=1

|λv+1||tv|k
m+1∑
n=v+1

(
Pn
pn

)δk
|∆v(ânv)|

= O(1)

m∑
v=1

(
Pv
pv

)δk−1
|λv+1||tv|k

= O(1) as m →∞,

by using (2.5), (2.6), (3.1), (3.2) and (3.3).



AN APPLICATION OF δ-QUASI MONOTONE SEQUENCE 139

Finally, as in In,1, we have that
m∑
n=1

(
Pn
pn

)δk+k−1
|In,4|k = O(1)

m∑
n=1

(
Pn
pn

)δk+k−1
|λn|k|tn|kaknn

= O(1)

m∑
n=1

(
Pn
pn

)δk−1
|λn||λn|k−1|tn|k

= O(1)

m∑
n=1

(
Pn
pn

)δk−1
|λn||tn|k

= O(1) as m →∞,
by using (2.5), (3.1), (3.2) and (3.3). This completes the proof of Theorem 3.1.

It should be noted that if we take δ = 0 in Theorem 3.1, then we get Theorem 2.2. In this case,
condition (3.2) reduces to condition (2.2). Also, if we take δ = 0 and anv =

pv
Pn

, then we get Theorem
2.1.

References

[1] R. P. Boas, Quasi-positive sequences and trigonometric series, Proc. London Math. Soc. 14A (1965), 38-46.

[2] H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), 147-149.
[3] H. Bor, On quasi-monotone sequences and their applications, Bull. Austral. Math. Soc. 43 (1991), 187-192.

[4] H. Bor, On local property of | N̄,pn; δ |k summability of factored Fourier series, J. Math. Anal. Appl. 179 (1993),
646–649.

[5] G. H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.

[6] H. S. Özarslan and H. N. Öğdük, Generalizations of two theorems on absolute summability methods, Aust. J. Math.
Anal. Appl. 1 (1) (2004), Article 13, 7 pp.

[7] H. S. Özarslan and M. Ö. Şakar, A new application of absolute matrix summability, Math. Sci. Appl. E-Notes 3
(2015), 36-43.

[8] 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.
[9] 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; hseyhan38@gmail.com


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