International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 1 (2016), 40-42
http://www.etamaths.com

A NEW RESULT ON GENERALIZED ABSOLUTE CESÀRO SUMMABILITY

HÜSEYIN BOR1,∗ AND RAM N. MOHAPATRA2

Abstract. In [4], a main theorem dealing with an application of almost increasing sequences, has

been proved. In this paper, we have extended that theorem by using a general class of quasi power

increasing sequences, which is a wider class of sequences, instead of an almost increasing sequence.
This theorem also includes some new and known results.

1. Introduction

A positive sequence (bn) is said to be an almost increasing sequence if there exists a positive increasing
sequence (cn) and two positive constants M and N such that Mcn ≤ bn ≤ Ncn (see [1]). 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 [2]). 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(σ,γ)} = {nσ(log n)γ, γ ≥ 0, 0 < σ < 1}(see [11]). If we take γ=0, then we get a quasi-σ-
power increasing sequence. Every almost increasing sequence is a quasi-σ-power increasing sequence for any
non-negative σ, but the converse is not true for σ > 0 (see [9]). Let

∑
an be a given infinite series. We denote

by tα,βn the nth Cesàro mean of order (α,β), with α + β > −1, of the sequence (nan), that is (see [6])

(1) t
α,β
n =

1

A
α+β
n

n∑
v=1

A
α−1
n−vA

β
vvav,

where

(2) A
α+β
n = O(n

α+β
), A

α+β
0 = 1 and A

α+β
−n = 0 for n > 0.

Let (θα,βn ) be a sequence defined by (see [3])

θ
α,β
n =

{ ∣∣tα,βn ∣∣ , α = 1,β > −1
max1≤v≤n

∣∣tα,βv ∣∣ , 0 < α < 1,β > −1.(3)
The series

∑
an is said to be summable | C,α,β |k, k ≥ 1, if (see [7])

(4)

∞∑
n=1

1

n
| tα,βn |

k
< ∞.

If we take β = 0, then | C,α,β |
k

summability reduces to | C,α |
k

summability (see [8]).
The first author has proved the following main theorem.
Theorem A ([4]). Let (θα,βn ) be a sequence defined as in (3). Let (Xn) be an almost increasing sequence
such that | ∆Xn |= O(Xn/n) and let λn → 0 as n → ∞. Suppose that there exists a sequence of numbers
(An) such that it is δ-quasi-monotone with

∑
nδnXn < ∞,

∑
AnXn is convergent, and | ∆λn |≤ | An | for all

n. If the condition
m∑
n=1

(θα,βn )
k

n
= O(Xm) as m →∞(5)

satisfies, then the series
∑
anλn is summable | C,α,β |k, 0 < α ≤ 1, α + β > 0, and k ≥ 1.

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G05, 40G99.
Key words and phrases. Cesàro mean; power increasing sequence; quasi-monotone sequence; summability factors;

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.

40



A NEW RESULT ON GENERALIZED ABSOLUTE CESÀRO SUMMABILITY 41

2. The main result.

The aim of this paper is to extent Theorem A by using a quasi-f-power increasing sequence, which is a
general class of quasi power increasing sequences, instead of an almost increasing sequence. We shall prove the
following theorem.
Theorem. Let (θα,βn ) be a sequence defined as in (3). Let (Xn) be a quasi-f-power increasing sequence and
let λn → 0 as n →∞. Suppose that there exists a sequence of numbers (An) such that it is δ-quasi-monotone
with ∆An ≤ δn,

∑
nδnXn < ∞,

∑
AnXn is convergent, and | ∆λn |≤ | An | for all n. If the condition (5) is

satisfied, then the series
∑
anλn is summable | C,α,β |k, 0 < α ≤ 1, α + β > 0, and k ≥ 1.

If we take (Xn) as an almost increasing sequence such that | ∆Xn |= O(Xn/n), then we get Theorem A, in
this case condition ’∆An ≤ δn’ is not needed.
We need the following lemmas for the proof of our theorem.
Lemma 1 ([3]). If 0 < α ≤ 1, β > −1, and 1 ≤ v ≤ n, then

(6) |
v∑
p=0

A
α−1
n−pA

β
pap |≤ max

1≤m≤v
|
m∑
p=0

A
α−1
m−pA

β
pap | .

Lemma 2 ([5]). Let (Xn) be a quasi-f-power increasing sequence. If (An) is a δ-quasi-monotone sequence
with ∆An ≤ δn and

∑
nδnXn < ∞ , then

∞∑
n=1

nXn | ∆An |< ∞,(7)

nAnXn = O(1) as n →∞.(8)

3. Proof of the theorem

Let (Tα,βn ) be the nth (C,α,β) mean of the sequence (nanλn). Then, by (1), we have

T
α,β
n =

1

A
α+β
n

n∑
v=1

A
α−1
n−vA

β
vvavλv.

Applying Abel’s transformation first and then using Lemma 1, we obtain that

T
α,β
n =

1

A
α+β
n

n−1∑
v=1

∆λv

v∑
p=1

A
α−1
n−pA

β
ppap +

λn

A
α+β
n

n∑
v=1

A
α−1
n−vA

β
vvav,

| Tα,βn | ≤
1

A
α+β
n

n−1∑
v=1

| ∆λv ||
v∑
p=1

A
α−1
n−pA

β
ppap | +

| λn |
A
α+β
n

|
n∑
v=1

A
α−1
n−vA

β
vvav |

≤
1

A
α+β
n

n−1∑
v=1

A
(α+β)
v θ

α,β
v | ∆λv | + | λn | θ

α,β
n = T

α,β
n,1 + T

α,β
n,2 .

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

∞∑
n=1

1

n
| Tα,βn,r |

k
< ∞, for r = 1, 2.



42 BOR AND MOHAPATRA

When k > 1, we can apply Hölder’s inequality with indices k and k′, where 1
k

+ 1
k′ = 1, we get that

m+1∑
n=2

1

n
| Tα,βn,1 |

k ≤
m+1∑
n=2

1

n
|

1

A
α+β
n

n−1∑
v=1

A
(α+β)
v θ

α,β
v ∆λv |

k

= O(1)

m+1∑
n=2

1

n1+(α+β)k

{
n−1∑
v=1

v
(α+β)k| Av |(θα,βv )

k

}
×

{
n−1∑
v=1

| Av |

}k−1

= O(1)

m∑
v=1

v
(α+β)k| Av |(θα,βv )

k
m+1∑
n=v+1

1

n1+(α+β)k

= O(1)

m∑
v=1

v
(α+β)k| Av |(θα,βv )

k

∫ ∞
v

dx

x1+(α+β)k
= O(1)

m∑
v=1

v| Av |
(θα,βv )

k

v

= O(1)

m−1∑
v=1

∆(v| Av |)
v∑
p=1

(θα,βp )
k

p
+ O(1)m| Am |

m∑
v=1

(θα,βv )
k

v

= O(1)

m−1∑
v=1

| (v + 1)∆ | Av | − | Av || Xv + O(1)m| Am |Xm

= O(1)

m−1∑
v=1

v | ∆Av | Xv + O(1)
m−1∑
v=1

| Av |Xv + O(1)m| Am |Xm

= O(1) as m →∞,
in view of hypotheses of the theorem and Lemma 2. Similarly, we have that

m∑
n=1

1

n
| Tα,βn,2 |

k
= O(1)

m∑
n=1

| λn |
n

(θ
α,β
n )

k
= O(1)

m∑
n=1

(θα,βn )
k

n

∞∑
v=n

| ∆λv |

= O(1)

∞∑
v=1

| ∆λv |
v∑

n=1

(θα,βn )
k

n
= O(1)

∞∑
v=1

| ∆λv | Xv

= O(1)

∞∑
v=1

| Av |Xv < ∞.

This completes the proof of the theorem. If we take β = 0, then we get a new result concerning the | C,α |
k

summability factors. If we set β = 0, α = 1, and Xn= logn, then we obtain the result of Mazhar dealing
with | C, 1 |

k
summability factors (see [10]). Finally, if we take γ=0, then we get a new result dealing with an

application of quasi-σ-power increasing sequences.

References

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[7] G. Das, A Tauberian theorem for absolute summability, Proc. Camb. Phil. Soc., 67 (1970), 321-326.
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[10] S. M. Mazhar, On generalized quasi-convex sequence and its applications, Indian J. Pure Appl. Math., 8 (1977),
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1224-1230.

1P. O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey

2University of Central Florida, Orlando, FL 32816, USA

∗Corresponding author: hbor33@gmail.com