International Journal of Analysis and Applications
ISSN 2291-8639
Volume 13, Number 1 (2017), 93-97
http://www.etamaths.com

SOME EQUIVALENCE THEOREMS ON ABSOLUTE SUMMABILITY METHODS

HİKMET SEYHAN ÖZARSLAN∗

Abstract. In this paper, we obtained necessary and sufficient conditions for the equivalence of two

general summability methods. Some new and known results are also obtained.

1. Introduction

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

tn =
1

Pn

n∑
v=0

pvsv (1.2)

defines the sequence (tn) of the Riesz mean or simply the (N̄,pn) mean of the sequence (sn), generated
by the sequence of coefficients (pn) (see [7]). The series

∑
an is said to be summable | N̄,pn; δ |k,

k ≥ 1 and δ ≥ 0, if (see [5])
∞∑
n=1

(
Pn
pn

)δk+k−1
| ∆tn−1 |k< ∞, (1.3)

where

∆tn−1 = −
pn

PnPn−1

n∑
v=1

Pv−1av, n ≥ 1. (1.4)

If we set δ = 0, then we obtain | N̄,pn |k summability (see [1]). In the special case pn = 1 for all
values of n, then | N̄,pn; δ |k summability is the same as | C, 1; δ |k summability (see [6]). Also if we
take δ = 0 and k = 1, then we get | N̄,pn | summability.
Let (ϕn) be any sequence of positive real numbers. The series

∑
an is summable ϕ−|N̄,pn; δ|k, k ≥ 1,

if (see [8])

∞∑
n=1

ϕδk+k−1n |∆tn−1|
k < ∞. (1.5)

If we take ϕn =
Pn
pn

, then ϕ−|N̄,pn; δ|k summability reduces to |N̄,pn; δ|k summability. Also, if we
take δ = 0 and ϕn =

Pn
pn

, then ϕ−|N̄,pn; δ|k summability reduces to |N̄,pn|k summability.

Received 22nd July, 2016; accepted 18th September, 2016; published 3rd January, 2017.
2010 Mathematics Subject Classification. 26D15, 40F05, 40G05, 40G99, 46A45.
Key words and phrases. Riesz mean; absolute summability; Hölder inequality; equivalence theorem; Minkowski

inequality; infinite series; 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.

93



94 ÖZARSLAN

2. Known Results

We say that two summability methods are equivalent if they sum the same set of series (not nec-
essarily to the same sums). Bor and Thorpe proved the following theorems about the | N̄,pn |k and
| N̄,qn |k summability methods.
Theorem 2.1 ( [2]). Let (pn) and (qn) be positive sequences and k ≥ 1. In order that | N̄,pn |k
should be equivalent to | N̄,qn |k it is sufficient that

qnPn
Qnpn

= O(1) (2.1)

and
pnQn
Pnqn

= O(1) (2.2)

hold.
Theorem 2.2 ( [4]). Let (pn) and (qn) be positive sequences and k ≥ 1. In order that every | N̄,pn |k
summable series be | N̄,qn |k summable it is necessary that (2.1) holds. If (2.2) holds then (2.1) is
also sufficient for the conclusion.
Theorem 2.3 ( [4]). Let (pn) and (qn) be positive sequences and k ≥ 1. In order that

∣∣N̄,pn∣∣k be
equivalent to | N̄,qn |k it is necessary and sufficient that (2.1) and (2.2) hold.

3. Main Results

The aim of this paper is to generalize Theorem 2.2 and Theorem 2.3 for the general summability
methods. Now, we shall prove the following theorems.
Theorem 3.1. Let k ≥ 1 and 0 ≤ δ < 1/k. (ϕn), (pn) and (qn) be sequences of positive numbers,
and let

m+1∑
n=v+1

ϕδk+k−1n q
k
n

QknQn−1
= O

{
ϕδk+k−1v

qk−1v
Qkv

}
as m →∞. (3.1)

In order that every ϕ− | N̄,pn; δ |k summable series be ϕ− | N̄,qn; δ |k summable it is necessary that
(2.1) holds. If (2.2) holds then (2.1) is also sufficient for the conclusion.

It should be noted that if we take ϕn =
Pn
pn

, δ = 0 for ϕ− | N̄,pn; δ |k and ϕn = Qnqn , δ = 0 for
ϕ− | N̄,qn; δ |k, then Theorem 3.1 reduces to Theorem 2.2. In this case condition (8) reduces to

m+1∑
n=v+1

qn
QnQn−1

= O

(
1

Qv

)
as m →∞, (3.2)

which always exists.
It is also remarked that if we take ϕn =

Qn
qn

and qn = 1 for all values of n, then the condition (8) fulfils.

We need the following lemma for the proof of Theorem 3.1.
Lemma 3.2 ( [3]). Let k ≥ 1 and A = (anv) be an infinite matrix. In order that A ∈ (lk, lk) it is
necessary that

anv = O(1) for all n,v ≥ 0. (3.3)

4. Proof of Theorem 3.1.

Firstly we prove sufficiency. Let (tn) denote (N̄,pn) mean of the series
∑
an. Then, by definition,

we have

tn =
1

Pn

n∑
v=0

pvsv =
1

Pn

n∑
v=0

(Pn −Pv−1)av. (4.1)

If the series
∑
an is summable ϕ−

∣∣N̄,pn; δ∣∣k, then
∞∑
n=1

ϕδk+k−1n |∆tn−1|
k
< ∞. (4.2)



SOME EQUIVALENCE THEOREMS ON ABSOLUTE SUMMABILITY METHODS 95

Since,

∆tn−1 =

(
−

1

Pn−1
+

1

Pn

) n∑
v=0

Pv−1av

= −
pn

PnPn−1

n∑
v=1

Pv−1av, n ≥ 1, (P−1 = 0), (4.3)

we have

Pn−1an = −
PnPn−1
pn

∆tn−1 +
Pn−1Pn−2
pn−1

∆tn−2. (4.4)

That is

an = −
Pn
pn

∆tn−1 +
Pn−2
pn−1

∆tn−2. (4.5)

If (Tn) denotes the (N̄,qn) mean of the series
∑
an, similarly we have that

Tn =
1

Qn

n∑
v=0

qvsv =
1

Qn

n∑
v=0

(Qn −Qv−1)av. (4.6)

Hence

∆Tn−1 = −
qn

QnQn−1

n∑
v=1

Qv−1av, n ≥ 1, (Q−1 = 0). (4.7)

Since

av = −
Pv
pv

∆tv−1 +
Pv−2
pv−1

∆tv−2,

by (15), we have that

∆Tn−1 = −
qn

QnQn−1

n∑
v=1

Qv−1

(
−
Pv
pv

∆tv−1 +
Pv−2
pv−1

∆tv−2

)

=
qnPn
pnQn

∆tn−1 +
qn

QnQn−1

n−1∑
v=1

Qv−1
Pv
pv

∆tv−1 −
qn

QnQn−1

n−1∑
v=1

Qv
Pv−1
pv

∆tv−1

=
qnPn
pnQn

∆tn−1 +
qn

QnQn−1

n−1∑
v=1

∆tv−1
pv

(Qv−1Pv −QvPv−1) .

Also,

Qv−1Pv −QvPv−1 = Qv−1Pv −Qv (Pv −pv) = Qv−1Pv −QvPv + pvQv

= (Qv−1 −Qv) Pv + pvQv = −qvPv + pvQv,

so that

∆Tn−1 =
qnPn
Qnpn

∆tn−1 −
qn

QnQn−1

n−1∑
v=1

Pv
pv
qv∆tv−1 +

qn
QnQn−1

n−1∑
v=1

Qv∆tv−1

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

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

∞∑
n=1

ϕδk+k−1n |Tn,r|
k
< ∞, for r = 1, 2, 3. (4.8)



96 ÖZARSLAN

Firstly, by using (6) and (12), we have
m∑
n=1

ϕδk+k−1n |Tn,1|
k

=

m∑
n=1

ϕδk+k−1n

∣∣∣∣ qnPnQnpn ∆tn−1
∣∣∣∣k

= O(1)

m∑
n=1

ϕδk+k−1n |∆tn−1|
k

= O(1) as m →∞.

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

+ 1
k′

= 1, we have that

m+1∑
n=2

ϕδk+k−1n |Tn,2|
k

=

m+1∑
n=2

ϕδk+k−1n

∣∣∣∣∣ qnQnQn−1
n−1∑
v=1

Pv
pv
qv∆tv−1

∣∣∣∣∣
k

≤
m+1∑
n=2

ϕδk+k−1n
qkn

QknQ
k
n−1

{
n−1∑
v=1

Pv
pv
qv |∆tv−1|

}k

≤
m+1∑
n=2

ϕδk+k−1n
qkn

QknQn−1

{
n−1∑
v=1

(
Pv
pv

)k
qv |∆tv−1|

k

}

×

{
1

Qn−1

n−1∑
v=1

qv

}k−1

= O(1)

m∑
v=1

(
Pv
pv

)k
qv |∆tv−1|

k
m+1∑
n=v+1

ϕδk+k−1n q
k
n

QknQn−1

= O(1)

m∑
v=1

(
Pv
pv

)k
qv |∆tv−1|

k
ϕδk+k−1v

qk−1v
Qkv

= O(1)

m∑
v=1

ϕδk+k−1v |∆tv−1|
k

= O(1) as m →∞,
by virtue of the hypotheses of Theorem 3.1.
Finally, as in Tn,2, we have that

m+1∑
n=2

ϕδk+k−1n |Tn,3|
k

=

m+1∑
n=2

ϕδk+k−1n

∣∣∣∣∣ qnQnQn−1
n−1∑
v=1

Qv∆tv−1

∣∣∣∣∣
k

≤
m+1∑
n=2

ϕδk+k−1n
qkn

QknQ
k
n−1

{
n−1∑
v=1

Qv
qv
qv |∆tv−1|

}k

≤
m+1∑
n=2

ϕδk+k−1n
qkn

QknQn−1

{
n−1∑
v=1

(
Qv
qv

)k
qv |∆tv−1|

k

}

×

{
1

Qn−1

n−1∑
v=1

qv

}k−1

= O(1)

m∑
v=1

(
Qv
qv

)k
qv |∆tv−1|

k
m+1∑
n=v+1

ϕδk+k−1n q
k
n

QknQn−1

= O(1)

m∑
v=1

(
Qv
qv

)k
qv |∆tv−1|

k
ϕδk+k−1v

qk−1v
Qkv

= O(1)

m∑
v=1

ϕδk+k−1v |∆tv−1|
k

= O(1) as m →∞,



SOME EQUIVALENCE THEOREMS ON ABSOLUTE SUMMABILITY METHODS 97

by virtue of the hypotheses of Theorem 3.1.
Therefore, we get

∞∑
n=1

ϕδk+k−1n |Tn,r|
k

= O(1) as m →∞, for r = 1, 2, 3.

This completes the proof of sufficiency of Theorem 3.1.
For the proof of the necessity, we consider the series to series version of (2) i.e. for n ≥ 1, let

bn =
pn

PnPn−1

n∑
v=1

Pv−1av, cn =
qn

QnQn−1

n∑
v=1

Qv−1av.

A simple calculation shows that for n ≥ 1

cn =
qn

QnQn−1

n−1∑
v=1

bv
pv

(Qv−1Pv −QvPv−1) +
qnPn
pnQn

bn.

From this we can write down at once the matrix A that transforms
(
ϕ
δk+k−1

k
n bn

)
into

(
ϕ
δk+k−1

k
n cn

)
.

Thus every ϕ−
∣∣N̄,pn; δ∣∣k summable series is ϕ− ∣∣N̄,qn; δ∣∣k summable if and only if A ∈ (lk, lk). By

Lemma 3.2, it is necessary that the diagonal terms of A must be bounded, which gives that (6) must
hold.
Theorem 3.2. Let (pn) and (qn) be positive sequences satisfying the condition (8), k ≥ 1 and
0 ≤ δ < 1/k. In order that ϕ− | N̄,pn; δ |k be equivalent to ϕ− | N̄,qn; δ |k it is necessary and
sufficient that (6) and (7) hold.

It should be remarked that if we set ϕn =
Pn
pn

, δ = 0 for ϕ− | N̄,pn; δ |k and ϕn = Qnqn , δ = 0 for
ϕ− | N̄,qn; δ |k, then Theorem 3.2 reduces to Theorem 2.3.
Proof of Theorem 3.2. Interchange the roles of (pn) and (qn) in Theorem 3.1.

References

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

[2] H. Bor and B. Thorpe, On some absolute summability methods, Analysis 7 (1987), 145–152.

[3] H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc. 113 (1991), 1009–
1012.

[4] H. Bor and B. Thorpe, A note on two absolute summability methods, Analysis 12 (1992), 1–3.
[5] H. Bor, On local property of | N̄,pn; δ |k summability of factored Fourier series, J. Math. Anal. Appl. 179 (2) (1993),

646–649.

[6] T. M. Flett, Some more theorems concerning the absolute summability of Fourier series and power series, Proc.
London Math. Soc. 8 (3) (1958), 357-387.

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

[8] H. Seyhan, On the local property of ϕ− | N̄,pn; δ |k summability of factored Fourier series, Bull. Inst. Math. Acad.
Sinica 25 (1997), 311-316.

Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey

∗Corresponding author: seyhan@erciyes.edu.tr; hseyhan38@gmail.com


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