Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON THE ABSOLUTE RIESZ SUMMABILITY FACTORS
L. LEINDLER
Bolyai Institute, University of Szeged
Aradi vértanúk tere 1
H-6720 Szeged, Hungary.
volume 6, issue 4, article 96,
2005.
Received 01 April, 2005;
accepted 12 August, 2005.
Communicated by: S.S. Dragomir
EMail: leindler@math.u-szeged.hu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
098-05
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Abstract
A crucial assumption of a previous theorem of the author is omitted without
changing the consequence. This is achieved by proving a new (?) estimation
on the absolute value of the terms of a real sequence by means of the sums of
the differences of the terms.
2000 Mathematics Subject Classification: 40A05, 40D15, 40F05.
Key words: Infinite Series, First and second differences, Riesz summability.
A Note on the Absolute Riesz
Summability Factors
This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant No. T042462 and TS44782.
L. Leindler
Title Page
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Contents
3
4
6
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1.
Introduction
In [4] we proved a theorem on absolute Riesz summability. Our paper was
initiated by a theorem of H. Bor [2] (see also [3]). Now we do not intend to
recall these theorems, the interested readers are referred to [4]. The aim of the
present note is to show that the crucial condition of our proof, λn → 0, can be
deduced from two other conditions of the theorem.
In order to provide the new theorem we require some notions and notations.
A positive sequence {an } is said to be quasi increasing if there exists a
constant K = K({ak }) ≥ 1 such that
(1.1)
K an ≥ am
holds for all n ≥ m.
P
The series ∞
n=1 an with partial sums sn is said to be summable |N , pn |k , k ≥
1, if
k−1
∞ X
Pn
|tn − tn−1 |k < ∞,
pn
n=1
where {pn } is a sequence of positive numbers such that
Pn :=
n
X
pν → ∞,
ν=0
and
n
1 X
p ν sν .
tn :=
Pn ν=0
A Note on the Absolute Riesz
Summability Factors
L. Leindler
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2.
Result
As we have written above, the new theorem to be presented here deviates from
our previous result merely that an assumption, λn → 0, does not appear among
the conditions.
The new theorem reads as follows.
Theorem 2.1. Let {λn } be a sequence of real numbers satisfying the condition
∞
X
1
|λn | < ∞.
n
n=1
(2.1)
Suppose that there exists a positive quasi increasing sequence {Xn } such that
(2.2)
∞
X
Xn |∆ λn | < ∞,
(∆ λn := λn − λn+1 ),
A Note on the Absolute Riesz
Summability Factors
L. Leindler
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n=1
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∗
Xm
:=
(2.3)
m
X
n=1
1
|tn |k = O(Xm ),
n
m
X
pn
|tn |k = O(Xm )
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n=1 n
(2.4)
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and
∞
X
(2.5)
n Xn∗ |∆(|∆ λn |)| < ∞
n=1
hold. Then the series
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an λn is summable |N , pn |k , k ≥ 1.
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It is clear that if we can verify first that the conditions (2.1) and (2.2) imply
that λn → 0, then the proof given in [4] is acceptable now, too. We shall follow
this way.
A Note on the Absolute Riesz
Summability Factors
L. Leindler
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3.
Lemmas
We need the following lemmas for the proof our statement.
Lemma 3.1 ([1, 2.2.2. p. 72]). If {µn } is a positive, monotone increasing
P
and tending to infinity sequence, then the convergence of the series
un µ−1
n
implies the estimate
n
X
(3.1)
uk = o(µn ).
A Note on the Absolute Riesz
Summability Factors
k=1
This lemma is the famous Kronecker lemma.
Lemma 3.2. Let {γn } be a sequence of real numbers and denote
Γn :=
n
X
γk
and
Rn :=
k=1
∞
X
|∆ γk |.
k=n
If Γn = o(n) then there exists a natural number n0 such that
(3.2)
|γn | ≤ 2Rn
for all n ≥ n0 . Naturally R1 < ∞ is assumed, otherwise (3.2) is a triviality.
However then Γn = o(n) is not only sufficient but also necessary to (3.2).
Remark 1. It is clear that if γn → 0 then |γn | ≤ Rn is trivial, but not if γn 6→ 0,
see e.g. γn = c 6= 0 or γn = 2 − n1 . Perhaps (3.2) is known, but unfortunately
I have not encountered it in any paper. I presume that (3.2) is not very known,
namely recently two papers used it without the assumption Γn = o(n), or its
consequences to be given next.
L. Leindler
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4.
Corollaries
Lemma 3.2 implies the following usable consequences.
Corollary 4.1. Let {ρn } be a sequence of real numbers. If ρn = o(n) then
(4.1)
|∆ ρn | ≤ 2
∞
X
|∆2 ρk |,
(∆2 ρk = ∆(∆ ρk )).
k=n
holds if n is large enough.
A Note on the Absolute Riesz
Summability Factors
Corollary 4.2. Let α ≥ 0 and {ρn } be as in Corollary 4.1. If
(4.2)
∞
X
k α |∆2 ρk | < ∞,
(ρn = o(n)),
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k=1
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then
(4.3)
|∆ ρn | = o(n−α ).
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In my view Lemma 3.2 and these corollaries are of independent interest.
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5.
Proofs
Proof of Lemma 3.2. Let us assume that (3.2) does not hold for any n0 . Then
there exists an increasing sequence {νn } of the natural numbers such that
(5.1)
2Rνn < |γνn |.
Let m = νn , and be fixed. Then for any k > m
k−1
X
2Rm < |γm | = ∆ γi + γk ≤ Rm + |γk |,
i=m
A Note on the Absolute Riesz
Summability Factors
L. Leindler
whence
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Rm < |γk |
(5.2)
holds.
Now let us choose n such that
(5.3)
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(n − m)Rm > 2|Γm |.
It is easy to verify that for all k > m the terms γk have the same sign, that
is, γk · γk+1 > 0. Namely if γk and γk+1 have different sign then, by (5.2),
|∆ γk | > 2Rm . But this contradicts the fact that Rm ≥ Rk ≥ |∆ γk |.
Thus, if n > m then
n
X
Γn = Γm +
γk ,
k=m+1
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and by invoking inequalities (5.2) and (5.3) we obtain that
|Γn | ≥
n
X
1
|γk | − |Γm | ≥ (n − m)Rm .
2
k=m+1
Since the last inequality opposes the assumption Γn = o(n), thus (3.2) is
proved. To verify the necessity of the condition Γn = o(n) it suffices to observe that R1 < ∞ implies Rn → 0, thus, by (3.2),
n
1X
|γk | → 0
n k=1
clearly holds.
A Note on the Absolute Riesz
Summability Factors
L. Leindler
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Proof of Corollary 4.1. Applying Lemma 3.2 with γn := ∆ ρn , we promptly
get the statement of Corollary 4.1.
Proof of Corollary 4.2. In view of (4.2) it is plain that
∞
X
|∆2 ρk | = o(n−α ),
k=n
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whence (4.3) follows by (4.1).
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Proof of Theorem 2.1. It is clearly sufficient to verify that the conditions (2.1)
and (2.2) imply that
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J. Ineq. Pure and Appl. Math. 6(4) Art. 96, 2005
(5.4)
λn → 0,
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namely with this additional condition the assertion of Theorem 2.1 had been
proved in [4].
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Now we prove (5.4). In view of Lemma 3.1 we know that nk=1 |λk | = o(n),
thus the assumptions of Lemma 3.2 are satisfied with γn := λn . Furthermore
the condition (2.2) visibly implies that
(5.5)
∞
X
|∆ λk | = o(1),
k=n
thus (3.2), by (5.5), proves (5.4).
The proof is complete.
A Note on the Absolute Riesz
Summability Factors
L. Leindler
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References
[1] G. ALEXITS, Convergence Problems of Orthogonal Series. Pergamon
Press 1961, ix+350 pp. (Orig. German ed. Berlin 1960.)
[2] H. BOR, An application of almost increasing and δ-quasi-monotone sequences, J. Inequal. Pure and Appl. Math., 1(2) (2000), Art. 18. [ONLINE:
http://jipam.vu.edu.au/article.phd?sid=112]
[3] H. BOR, Corrigendum on the paper “An application of almost increasing
and δ-quasi-monotone sequences", J. Inequal. Pure and Appl. Math., 3(1)
(2002), Art. 16. [ONLINE: http://jipam.vu.edu.au/article.
phd?sid=168]
[4] L. LEINDLER, On the absolute Riesz summability factors, J. Inequal. Pure
and Appl. Math., 5(2) (2004), Art. 29. [ONLINE: http://jipam.vu.
edu.au/article.phd?sid=376]
A Note on the Absolute Riesz
Summability Factors
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