ON EQUIVALENCE OF COEFFICIENT
CONDITIONS. II
Equivalence of Coefficient
Conditions
L. LEINDLER
Bolyai Institute, University of Szeged
Aradi vértanúk tere 1,
H-6720 Szeged, Hungary
EMail: leindler@math.u-szeged.hu
L. Leindler
vol. 9, iss. 3, art. 83, 2008
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Contents
Received:
28 August, 2008
Accepted:
16 September, 2008
Communicated by:
H. Bor
2000 AMS Sub. Class.:
26D15, 40D15.
Key words:
Numerical series, equiconvergence.
Abstract:
An additional theorem is proved pertaining to the equiconvergence of numerical
series.
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Contents
1
Introduction
3
2
Result
5
3
Lemma
6
4
Proof of Theorem 2.1
9
Equivalence of Coefficient
Conditions
L. Leindler
vol. 9, iss. 3, art. 83, 2008
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1.
Introduction
In the papers [2], [3] and [4] we have studied the relations of the following sums:
∞
X
S1 :=
cqn µn ,
n=1
S2 :=
S3 :=
∞
X
λn
n=1
k=n
∞
X
n
X
λn
n=1
S4 :=
∞
X
∞
X
n=1
! pq
cqk
S2∗ :=
,
! pq
cqk
S3∗ :=
,
νn+1 −1
λn
λn µ−1
n
!
cqk
p
q
,
S4∗
k=1
∞
X
∞
X
:=
∞
X
n=1
k=νn
n
X
n=1
λn µ−1
n
n=1
k=1
X
∞
X
Equivalence of Coefficient
Conditions
p
! q−p
λk
,
vol. 9, iss. 3, art. 83, 2008
p
! q−p
λk
,
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k=n
λn
λn
µν n
p
q−p
Contents
,
where 0 < p < q, λ := {λn } and c := {cn } are sequences of nonnegative numbers, ν := {νm } is a subsequence of natural numbers, and µ := {µn } is a certain
nondecreasing sequence of positive numbers.
In [2] we verified that S2 < ∞ if and only if there exists a µ satisfying the
conditions S1 < ∞ and S2∗ < ∞. Similarly S3 < ∞ if and only if S1 < ∞ and
S3∗ < ∞.
In [3] we showed that S4 < ∞ if and only if there exists a µ such that S1 < ∞
and S4∗ < ∞.
Recently, in [4], we proved that if
!1/q
∞
n
X
X
q
(1) p−q
(1)
µn := Λn Cn , where Cn :=
ck
and Λn :=
λk ,
k=n
L. Leindler
k=1
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then the sums S1 , S2 and S2∗ are already equiconvergent.
Furthermore if
!1/q
n
∞
X
X
q
(2)
(2) p−q
ck
and Λn :=
λk ,
µn := Λn C̃n , where C̃n :=
k=1
k=n
then the sums S1 , S3 and S3∗ are equiconvergent.
Comparing the results proved in [4] and that of [2] and [3], we can observe that
in the former one the explicit sequences {µn } are determined, herewith they state
more than the outcomes of [2] and [3], where only the existence of a sequence {µn }
is proved.
Furthermore, in [4] the equiconvergence of these concrete sums are guaranteed,
too.
However the equiconvergence in [4] is proved only in connection with the sums
S2 and S3 , but not for S4 . This is a gap or shortcoming at these investigations.
The aim of this note is closing this gap. Unfortunately we cannot give a complete
solution, namely our result to be verified requires an additional assumption on the
sequence λ. In particular, λ should be quasi geometrically increasing, that is, we
assume that there exist a natural number N and K ≥ 1 such that λn+N ≥ 2λn and
λn ≤ Kλn+1 hold for all n.
Then we can give an explicit sequence µ such that the sums S1 , S4 and S4∗ are
already equiconvergent. We also show that without some additional requirement on
λ the equiconvergence does not hold. See the last part. Thus the following open
problem can be raised: What is the weakest additional assumption on sequence λ
which ensures the equiconvergence of these sums?
Equivalence of Coefficient
Conditions
L. Leindler
vol. 9, iss. 3, art. 83, 2008
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2.
Result
Theorem 2.1. If 0 < p < q, c := {cn } is a sequence of nonnegative numbers, ν :=
{νm } is a subsequence of natural numbers, and λ := {λn } is a quasi geometrically
increasing sequence, and for νm ≤ n < νm+1
! pq −1
∞
X
µn := λm
cqk
, m = 0, 1, . . . ,
Equivalence of Coefficient
Conditions
k=νm
then the sums S1 , S4 and S4∗ are equiconvergent.
L. Leindler
vol. 9, iss. 3, art. 83, 2008
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3.
Lemma
In order to verify our theorem, first we shall prove a lemma regarding the equiconvergence of two special series.
Lemma 3.1. Let 0 < α < 1, a := {an } be a sequence of nonnegative numbers, ν :=
{νm } be a subsequence of natural numbers, and
P κ := {κm } be a quasi geometrically
increasing sequence. Furthermore let Ak := ∞
n=k an , and for νm ≤ n < νm+1 let
µn := κm Aα−1
νm ,
m = 0, 1, . . . .
Equivalence of Coefficient
Conditions
L. Leindler
vol. 9, iss. 3, art. 83, 2008
Then
(3.1)
σ1 :=
∞
X
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an µ n < ∞
Contents
n=1
holds if and only if
(3.2)
σ2 :=
∞
X
κm Aανm
< ∞.
m=1
Proof of Lemma 3.1. Before starting the proofs we note that the following inequality
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(3.3)
m
X
κn ≤ K κm
n=1
holds for all m, subsequent to the fact that κ is a quasi geometrically increasing
sequence (see e.g. [1, Lemma 1]). Here and later on K denotes a constant that is
independent of the parameters.
Close
Furthermore we verify a useful inequality. If 0 ≤ a < b, 0 < α < 1 and
b α − aα
= α ξ α−1 ,
b−a
(3.4)
then
ξ ≥ α1/(1−α) b =: ξ0 ,
namely if a = 0 then ξ = ξ0 . Hence we get that
Equivalence of Coefficient
Conditions
α ξ α−1 ≤ bα−1 .
(3.5)
L. Leindler
Now we show that (3.1) implies (3.2). Since An & 0, thus, by (3.3),
∞
X
κm Aανm
=
m=1
∞
X
κm
m=1
=
∞
X
∞
X
(Aανn − Aανn+1 )
≤K
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n=m
(Aανn − Aανn+1 )
n=1
(3.6)
vol. 9, iss. 3, art. 83, 2008
n
X
Contents
κm
m=1
∞
X
κn (Aανn
−
Aανn+1 ).
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n=1
Using the relations (3.4) and (3.5) we obtain that
!
νn+1 −1
X
Aανn − Aανn+1 =
ak α ξ α−1 ≤
k=νn
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νn+1 −1
X
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!
ak
Aα−1
νn .
k=νn
This and (3.6) yield that
∞
X
m=1
κm Aανm
≤K
∞
X
n=1
νn+1 −1
κn Aα−1
νn
X
k=νn
ak = K
∞ νn+1
X
X−1
n=1 k=νn
ak µ k .
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Herewith the implication (3.1) ⇒ (3.2) is proved.
The proof of (3.2) ⇒ (3.1) is very easy. Namely
∞
X
an µ n =
n=ν1
∞ νm+1
X
X−1
an µ n
m=1 n=νm
=
≤
∞
X
m=1
∞
X
νm+1 −1
κm Aα−1
νm
X
an
Equivalence of Coefficient
Conditions
n=νm
κm Aανm ,
L. Leindler
vol. 9, iss. 3, art. 83, 2008
m=1
that is, (3.2) ⇒ (3.1) is verified.
Thus the proof is complete.
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4.
Proof of Theorem 2.1
We shall use the result of Lemma 3.1 with α = pq , an = cqn and κm = λm . Then
P
q
An = ∞
k=n ck and for νm ≤ n < νm+1
∞
X
µn = µν m = λ m
(4.1)
! p−q
q
cqk
.
Equivalence of Coefficient
Conditions
k=νm
Then σ1 = S1 , thus by Lemma 3.1, S1 < ∞ implies that σ2 < ∞, that is,
S4 =
(4.2)
∞
X
νm+1 −1
λm
m=1
X
!
cqn
p
q
≤
n=νm
∞
X
λm
m=1
∞
X
!
S4∗
=
∞
X
n=1
λn

k=νn
cqk
p
 q−p
! q−p
q 

=
vol. 9, iss. 3, art. 83, 2008
p
q
cqn
= σ2 .
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n=νm
Contents
Moreover, by (4.1),

∞
 X
L. Leindler
∞
X
n=1
λn
∞
X
! pq
cqk
= σ2 ,
k=νn
thus S1 < ∞ implies that both S4 < ∞ and S4∗ < ∞ hold.
Conversely, if S4 < ∞, then it suffices to show that σ2 = S4∗ < ∞ also holds.
Applying the inequality
X α X
ak ≤
aαk , 0 < α ≤ 1, ak ≥ 0,
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and (3.3), we obtain that
σ2 =
∞
X
λm Ap/q
νm ≤
m=1
=
∞
X
! pq
νn+1 −1
X
cqk
n
X
n=1
n=m
k=νn
X
! pq
cqk
λm
X
Equivalence of Coefficient
Conditions
! pq
νn+1 −1
λn
νn+1 −1
m=1
k=νn
∞
X
∞
X
λm
m=1
n=1
≤K
∞
X
cqk
L. Leindler
vol. 9, iss. 3, art. 83, 2008
k=νn
= KS4 < ∞.
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This, (4.2) and, by Lemma 3.1, the implication σ2 < ∞ ⇒ σ1 = S1 < ∞
complete the proof of Theorem 2.1.
Proof of the necessity of some additional assumption on λ. Let p = 1, q = 2, λn =
log n, νn = n and
m−3 if n = 2m ,
cn :=
0
otherwise.
Then
S4 =
∞
X
log 2m
< ∞,
3
m
m=2
S4∗
but S1 < ∞ and
< ∞ cannot be fulfilled simultaneously. Namely, then with a
nondecreasing sequence {µn } the conditions
S1 =
∞
X
m=1
m−6 µ2m < ∞
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and
S4∗
∞
X
log2 m
=
<∞
µm
m=2
yield a trivial contradiction.
Equivalence of Coefficient
Conditions
L. Leindler
vol. 9, iss. 3, art. 83, 2008
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References
[1] L. LEINDLER, On equivalence of coefficient conditions with applications, Acta
Sci. Math. (Szeged), 60 (1995), 495–514.
[2] L. LEINDLER, On equivalence of coefficient conditions and application, Math.
Inequal. Appl. (Zagreb), 1 (1998), 41–51.
[3] L. LEINDLER, On a new equivalence of coefficient conditions and applications,
Math. Inequal. Appl., 10(2) (1999), 195–202.
Equivalence of Coefficient
Conditions
L. Leindler
vol. 9, iss. 3, art. 83, 2008
[4] L. LEINDLER, On equivalence of coefficient conditions, J. Inequal. Pure and
Appl. Math., 8(1) (2007), Art. 8. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=821].
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