International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 2 (2016), 95-100
http://www.etamaths.com

ON QUASI-POWER INCREASING SEQUENCES AND THEIR SOME

APPLICATIONS

HÜSEYIN BOR∗

Abstract. In [6], we proved a main theorem dealing with | N̄,pn,θn |k summability factors using
a new general class of power increasing sequences instead of a quasi-σ-power increasing sequence. In
this paper, we prove that theorem under weaker conditions. This theorem also includes some new

results.

1. Introduction

A positive sequence X = (Xn) is said to be a quasi-f-power increasing sequence if there exists
a constant K = K(X,f) ≥ 1 such that KfnXn ≥ fmXm for all n ≥ m ≥ 1, where f = (fn) =
{nσ(log n)η, η ≥ 0, 0 < σ < 1} (see [13]). If we set η=0, then we get a quasi-σ-power increas-
ing sequence (see [10]). We write BVO = BV ∩ CO, where CO = { x = (xk) ∈ Ω : limk |xk| = 0 },
BV ={ x = (xk) ∈ Ω :

∑
k |xk −xk+1| < ∞ } and Ω being the space of all real-valued sequences. Let∑

an be a given infinite series with the sequence of 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 [7]),

uαn =
1

Aαn

n∑
v=0

Aα−1n−vsv(1)

where

(2) Aαn =
(α + 1)(α + 2)....(α + n)

n!
= O(nα), Aα−n = 0 for n > 0.

A series
∑
an is said to be summable | C,α |k, k ≥ 1, if (see [8])

(3)

∞∑
n=1

nk−1 | uαn −u
α
n−1 |

k< ∞.

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

(4) Pn =
n∑
v=0

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

The sequence-to-sequence transformation

(5) vn =
1

Pn

n∑
v=0

pvsv

defines the sequence (vn) of the Riesz mean or simply the (N̄,pn) mean of the sequence (sn), generated
by the sequence of coefficients (pn) (see [9]). Let (θn) be any sequence of positive constants. The series∑
an is said to be summable | N̄,pn,θn |k,k ≥ 1, if (see [12])

∞∑
n=1

θk−1n | vn −vn−1 |
k
< ∞.(6)

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99, 46A45.
Key words and phrases. sequence spaces; Riesz mean; summability factors; increasing sequences; infinite series;

Hölder inequality; 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.

95



96 HÜSEYIN BOR

If we take θn =
Pn
pn
, then | N̄,pn,θn |k summability reduces to | N̄,pn |k summability (see [1]). Also,

if we take θn =
Pn
pn

and pn = 1 for all values of n, then we get | C, 1 |k summability. Furthermore, if
we take θn = n, then | N̄,pn,θn |k summability reduces to | R,pn |k summability (see [2]).
2. Known Results. The following theorems are known:

Theorem A ([4]). Let
(
θnpn
Pn

)
be a non-increasing sequence. Let (λn) ∈ BVO and let (Xn) be a

quasi- σ-power increasing sequence for some σ (0 < σ < 1) . Suppose also that there exist sequences
(βn) and (λn) such that

(7) | ∆λn |≤ βn,

(8) βn → 0 as n →∞,

(9)

∞∑
n=1

n | ∆βn | Xn < ∞,

(10) | λn | Xn = O(1).

If
n∑
v=1

θk−1v
| sv |k

vk
= O(Xn) as n →∞,(11)

and (pn) is a sequence such that

(12) Pn = O(npn),

(13) Pn∆pn = O(pnpn+1),

then the series
∑∞
n=1 an

Pnλn
npn

is summable | N̄,pn,θn |k, k ≥ 1.
Remark. We can take (λn) ∈BV instead of (λn) ∈BVO and it is sufficient to prove Theorem A.
Theorem B ([6]). Let

(
θnpn
Pn

)
be a non-increasing sequence. Let (λn) ∈ BV and let (Xn) be a

quasi-f-power increasing sequence for some σ (0 < σ < 1) and η ≥ 0. If the conditions (7)-(13) are
satisfied, then the series

∑∞
n=1 an

Pnλn
npn

is summable | N̄,pn,θn |k, k ≥ 1.
It should be noted that if we take η=0, then we obtain Theorem A.
3. The Main result. The purpose of this paper is to prove Theorem B under weaker conditions.
Now, we shall prove the following general theorem.

Theorem. Let
(
θnpn
Pn

)
be a non-increasing sequence. Let (Xn) be a quasi-f-power increasing sequence

for some σ (0 < σ < 1) and η ≥ 0. If the conditions (7)-(10), (12)-(13), and
n∑
v=1

θk−1v
| sv |k

vkXv
k−1 = O(Xn) as n →∞(14)

are satisfied, then the series
∑∞
n=1 an

Pnλn
npn

is summable | N̄,pn,θn |k, k ≥ 1.
Remark. It should be noted that condition (14) is reduced to the condition (11), when k=1. When
k > 1, the condition (14) is weaker than the condition (11), but the converse is not true. As in [14]
we can show that if (11) is satisfied, then we get that

n∑
v=1

θk−1v
| sv |k

vkXv
k−1 = O(

1

Xk−11
)

n∑
v=1

θk−1v
| sv |k

vk
= O(Xn).

If (14) is satisfied, then for k > 1 we obtain that

n∑
v=1

θk−1v
| sv |k

vk
=

n∑
v=1

θk−1v X
k−1
v

| sv |k

vkXv
k−1 = O(X

k−1
n )

n∑
v=1

θk−1v
| sv |k

vkXv
k−1 = O(X

k
n) 6= O(Xn).

Also, it should be noted that the condition ”(λn) ∈BV ” has been removed.
We require the following lemmas for the proof of the theorem.



QUASI-POWER INCREASING SEQUENCES 97

Lemma 1 ([5]). Under the conditions on (Xn), (βn) and (λn) as expressed in the statement of the
theorem, we have the following;

(15) nXnβn = O(1),

(16)

∞∑
n=1

βnXn < ∞.

Lemma 2 ([11]). If the conditions (12) and (13) are satisfied, then we have that

∆

(
Pn
npn

)
= O

(
1

n

)
.(17)

4. Proof of the theorem. 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, for n ≥ 1 we obtain that

Tn −Tn−1 =
pn

PnPn−1

n∑
v=1

Pv−1Pvavλv
vpv

.

Using Abel’s transformation, we get

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 prove the theorem, by Minkowski’s inequality, it is sufficient to show that
∞∑
n=1

θk−1n | Tn,r |
k< ∞, for r = 1, 2, 3, 4.(18)

Firstly, by using Abel’s transformation, we have that
m∑
n=1

θk−1n | Tn,1 |
k =

m∑
n=1

θk−1n n
−k | λn |k−1| λn || sn |k

= O(1)

m∑
n=1

| λn |
(

1

Xn

)k−1
θk−1n n

−k | sn |k

= O(1)

m−1∑
n=1

∆ | λn |
n∑
v=1

θk−1v
| sv |k

Xv
k−1vk

+ O(1) | λm |
m∑
n=1

θk−1n
| sn |k

Xn
k−1nk

= 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 →∞



98 HÜSEYIN BOR

by virtue of the hypotheses of the theorem and Lemma 1. Now, using (12) and applying Hölder’s
inequality, we have that

m+1∑
n=2

θk−1n | Tn,2 |
k = O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pkn−1
|
n−1∑
v=1

Pvsv∆λv |k

= O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pkn−1

{
n−1∑
v=1

Pv
pv
| sv | pv | ∆λv |

}k

= O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pn−1

n−1∑
v=1

(
Pv
pv

)k
| sv |k pv (βv)k

×

(
1

Pn−1

n−1∑
v=1

pv

)k−1

= O(1)

m∑
v=1

(
Pv
pv

)k
| sv |k pv (βv)k

m+1∑
n=v+1

(
θnpn
Pn

)k−1
pn

PnPn−1

= O(1)

m∑
v=1

(
Pv
pv

)k
| sv |k pv (βv)k

(
θvpv
Pv

)k−1 m+1∑
n=v+1

pn
PnPn−1

= O(1)

m∑
v=1

(
Pv
pv

)k
| sv |k (βv)k

(
pv
Pv

)
θk−1v

(
pv
Pv

)k−1
= O(1)

m∑
v=1

(vβv)
k−1vβv

1

vk
θk−1v | sv |

k

= O(1)

m∑
v=1

(
1

Xv

)k−1
vβv

1

vk
θk−1v | sv |

k

= O(1)

m−1∑
v=1

∆(vβv)

v∑
r=1

θk−1r
| sr |k

rkXr
k−1 + O(1)mβm

m∑
v=1

θk−1v
| sv |k

vkXv
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

βvXv + O(1)mβmXm = O(1)

as m →∞, in view of the hypotheses of the theorem and Lemma 1. Again, as in Tn,1, we have that

m+1∑
n=2

θk−1n | Tn,3 |
k = O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pkn−1

{
n−1∑
v=1

Pv | sv || λv |
1

v

}k

= O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pn−1

n−1∑
v=1

(
Pv
pv

)k
v−kpv | sv |k| λv |k

×

{
1

Pn−1

n−1∑
v=1

pv

}k−1



QUASI-POWER INCREASING SEQUENCES 99

= O(1)

m∑
v=1

(
Pv
pv

)k
v−k | sv |k pv | λv |k

m+1∑
n=v+1

(
θnpn
Pn

)k−1
pn

PnPn−1

= O(1)

m∑
v=1

(
Pv
pv

)k−1
v−kθk−1v

(
pv
Pv

)k−1
| λv |k−1| λv || sv |k

= O(1)

m∑
v=1

| λv |
(

1

Xv

)k−1
θk−1v v

−k | sv |k

= O(1)

m∑
v=1

| λv | θk−1v
| sv |k

vkXv
k−1 = O(1) as m →∞,

in view of the hypotheses of the theorem, Lemma 1 and Lemma 2. Finally, using Hölder’s inequality,
as in Tn,1 we have that

m+1∑
n=2

θk−1n | Tn,4 |
k =

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pkn−1
|
n−1∑
v=1

sv
Pv
v
λv |k

= O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pkn−1
|
n−1∑
v=1

sv
Pv
vpv

pvλ |k

= O(1)

m+1∑
n=2

θk−1n

(
pn
Pn

)k
1

Pn−1

n−1∑
v=1

| sv |k
(
Pv
pv

)k
v−kpv | λv |k

×

(
1

Pn−1

n−1∑
v=1

pv

)k−1

= O(1)

m∑
v=1

(
Pv
pv

)k
v−k | sv |k pv | λv |k

1

Pv

(
θvpv
Pv

)k−1
= O(1)

m∑
v=1

(
Pv
pv

)k−1
v−k

(
pv
Pv

)k−1
θk−1v | λv |

k−1| λv || sv |k

= O(1)

m∑
v=1

| λv | θk−1v
| sv |k

vkXv
k−1 = O(1) as m →∞.

This completes the proof of the theorem. If we set η ≥ 0, then we obtain Theorem B under weaker
conditions. If we take pn = 1 for all values of n, then we have a new result for | C, 1,θn |k summability.
Furthermore, if we take θn = n, then we have another new result for | R,pn |k summability. Finally,
if we take pn = 1 for all values of n and θn = n, then we get a new result dealing with | C, 1 |k
summability factors.

References

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1012.

[3] H. Bor, A general note on increasing sequences, JIPAM. J. Inequal. Pure Appl. Math., 8 (2007), Article ID 82.
[4] H. Bor, New application of power increasing sequences, Math. Aeterna, 2 (2012), 423-429.
[5] H. Bor, A new application of generalized power increasing sequences, Filomat, 26 (2012), 631-635.

[6] H. Bor, A new application of generalized quasi-power increasing sequences, Ukrainian Math. J., 64 (2012), 731-738.

[7] E. Cesàro, Sur la multiplication des séries, Bull. Sci. Math., 14 (1890), 114-120.
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[11] K. N. Mishra and R. S. L. Srivastava, On | N̄,pn | summability factors of infinite series, Indian J. Pure Appl. Math.,
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100 HÜSEYIN BOR

[13] W. T. Sulaiman, Extension on absolute summability factors of infinite series, J. Math. Anal. Appl., 322 (2006),
1224-1230.

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

∗Corresponding author: hbor33@gmail.com