R E S E A R C H
Open Access
On generalized absolute Cesàro summability
factors
A Nihal Tuncer
**Correspondence:
Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey
Abstract
In this paper, a known theorem dealing with|C,
α
,γ
;δ
|ksummability factors has been generalized for|C,α
,β
,γ
;δ
|ksummability factors. Some results have also been obtained.MSC: 40D15; 40F05; 40G99
Keywords: Hölder’s inequality; quasi-monotone sequence; summability factors
1 Introduction
A sequence (bn) of positive numbers is said to be quasi-monotone ifnbn≥–ρbnfor
someρ> and is said to beδ-quasi-monotone, ifbn→,bn> ultimately andbn≥–δn,
where (δn) is a sequence of positive numbers (see []). Let
anbe a given infinite series
with partial sums (sn). We denote byuα,βn andtα,βn thenth Cesàro means of order (α,β),
withα+β> –, of the sequences (sn) and (nan), respectively, that is (see []),
uα,βn = Aα+βn
n
v=
Aα–n–vAβvsv, ()
tnα,β= Aα+βn
n
v=
Aα–n–vAβvvav, ()
where
Aα+βn =Onα+β, α+β> –, Aα+β = and Aα+β–n = forn> . ()
The seriesanis said to be summable|C,α,β|k,k≥ andα+β> –, if (see [])
∞
n=
nk–uα,βn –uα,βn–k<∞. ()
Sincetα,βn =n(uα,βn –uα,βn–) (see []), condition () can also be written as
∞
n= nt
α,β
n k
<∞. ()
The seriesanis said to be summable|C,α,β,γ;δ|k,k≥,α+β> –,δ≥ andγ is a
real number, if (see [])
∞
n=
nγ(δk+k–)uα,βn –uα,βn–k=
∞
n=
nγ(δk+k–)–ktnα,βk<∞. ()
If we takeβ= , then|C,α,β,γ;δ|ksummability reduces to|C,α,γ;δ|ksummability (see []).
2 Known result
In [], we have proved the following theorem dealing with|C,α,γ;δ|ksummability factors of infinite series.
Theorem A Let k≥, ≤δ<α≤,andγ be a real number such that–γ(δk+k– ) + (α+ )k> .Suppose that there exists a sequence of numbers(Bn)such that it isδ
-quasi-monotone with|λn| ≤ |Bn|,λn→as n→ ∞, ∞
n=nδnlogn<∞and ∞
n=nBnlogn is
convergent.If the sequence(wα
n)defined by(see[])
wα
n=t
α
n, α= , ()
wαn=max
≤v≤nt
α
v, <α< , ()
satisfies the condition
m
n=
nγ(δk+k–)–kwnαk=O(logm) as m→ ∞, ()
then the seriesanλnis summable|C,α,γ;δ|k.
3 The main result
The aim of this paper is to generalize Theorem A for|C,α,β,γ;δ|ksummability. We shall prove the following theorem.
Theorem Let k≥, ≤δ<α≤,andγ be a real number such that(α+β+ –γ(δ+ ))k> ,and let there be sequences(Bn)and(λn)such that the conditions of TheoremAare
satisfied.If the sequence(wα,βn )defined by
wα,βn =tnα,β, α= ,β> –, ()
wα,βn =max
≤v≤n
tvα,β, <α< ,β> –, ()
satisfies the condition
m
n=
nγ(δk+k–)–kwα,β
n k
=O(logm) as m→ ∞, ()
then the seriesanλnis summable|C,α,β,γ;δ|k.It should be noted that if we takeβ= ,
We need the following lemmas for the proof of our theorem.
Lemma ([]) Under the conditions on(Bn),as taken in the statement of the theorem,we
have the following:
nBnlogn=O(), ()
∞
n=
nlogn|Bn|<∞. ()
Lemma ([]) If <α≤,β> –,and≤v≤n,then
v p=
Aα–n–pAβpap ≤max≤m≤v
m p=
Aα–m–pAβpap
. ()
4 Proof of the theorem
Let (Tnα,β) be thenth (C,α,β) mean of the sequence (nanλn). Then by (), we have
Tnα,β= Aα+βn
n
v= Aα–n–vAβ
vvavλv.
Firstly applying Abel’s transformation and then using Lemma , we have that
Tnα,β= Aα+βn
n– v= λv v p= Aα–n–pAβ
ppap+
λn
Aα+βn n
v= Aα–n–vAβ
vvav,
Tnα,β≤ Aα+βn
n–
v=
|λv|
v p=
Aα–n–pAβppap +
|λn|
Aα+βn n v=
Aα–n–vAβvvav
≤
Aα+βn n–
v=
AαvAβvwα,βv |λv|+|λn|wα,βn =Tnα,β, +Tnα,β,, say
since
Tnα,β, +Tnα,β,k≤kTnα,β,k+Tnα,β,k. ()
In order to complete the proof of the theorem, by (), it is sufficient to show that forr= , ,
∞
n=
nγ(δk+k–)–kTnα,β,rk<∞.
Wheneverk> , we can apply Hölder’s inequality with indiceskandk, wherek+k = ,
we get that
m+
n=
nγ(δk+k–)–kTnα,β,k
≤ m+
n=
nγ(δk+k–)–k
Aα+βn
n–
v= Aα
=O()
m+
n=
n(α+β+–γ(δ+))k
n–
v=
vαkvβk|λv|wα,βv k
n–
v=
|λv|
k–
=O()
m+
n=
n(α+β+–γ(δ+))k
n–
v=
vαkvβk|Bv|wα,βv k
n–
v=
|Bv|
k–
=O()
m
v=
v(α+β)k|Bv|wα,βv k
m+
n=v+ n(α+β+–γ(δ+))k
=O()
m
v=
v(α+β)k|Bv|wα,βv k
∞
v
dx x(α+β+–γ(δ+))k
=O()
m
v=
|Bv|vγ(δk+k–)–k+wα,βv k
=O()
m
v=
v|Bv|vγ(δk+k–)–kwα,βv k
=O()
m–
v=
v|Bv|
v
p=
pγ(δk+k–)–kwα,β
p k
+O()m|Bm|
m
v=
vγ(δk+k–)–kwα,βv k
=O()
m–
v=
v|Bv|logv+O()m|Bm|logm
=O()
m–
v=
v|Bv|logv+O()
m–
v=
|Bv+|logv+O()m|Bm|logm
=O() asm→ ∞,
in view of the hypotheses of the theorem and Lemma . Similarly, we have that
m+
n=
nγ(δk+k–)–kTnα,β,k=O()
m
n=
|λn|nγ(δk+k–)–kwα,βn k
=O()
m–
n=
|λn|
n
v=
vγ(δk+k–)–kwα,βv k
+O()|λm|
m
v=
vγ(δk+k–)–kwα,β
v k
=O()
m–
n=
|λn|logn+O()|λm|logm
=O()
m–
n=
|Bn|logn+O()|λm|logm
=O() asm→ ∞,
by virtue of the hypotheses of the theorem and Lemma . Therefore, by (), we get that for r= , ,
∞
n=
nγ(δk+k–)–kTα,β
n,r k
This completes the proof of the theorem. If we takeδ= andγ = , then we get a result for|C,α,β|k summability factors. Also, if we takeβ= ,δ= , andα= , then we get a result for|C, |ksummability.
Competing interests
The author declares that they have no competing interests.
Received: 23 December 2011 Accepted: 17 October 2012 Published: 30 October 2012 References
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doi:10.1186/1029-242X-2012-253