Volume 9 (2008), Issue 3, Article 83, 5 pp.
ON EQUIVALENCE OF COEFFICIENT CONDITIONS. II
L. LEINDLER
B OLYAI I NSTITUTE
U NIVERSITY OF S ZEGED
A RADI VÉRTANÚK TERE 1
H-6720 S ZEGED , H UNGARY
leindler@math.u-szeged.hu
Received 28 August, 2008; accepted 16 September, 2008
Communicated by H. Bor
A BSTRACT. An additional theorem is proved pertaining to the equiconvergence of numerical
series.
Key words and phrases: Numerical series, equiconvergence.
2000 Mathematics Subject Classification. 26D15, 40D15.
1. I NTRODUCTION
In the papers [2], [3] and [4] we have studied the relations of the following sums:
∞
X
S1 :=
cqn µn ,
n=1
S2 :=
S3 :=
S4 :=
∞
X
λn
∞
X
n=1
k=n
∞
X
n
X
λn
!
cqk
cqk
k=1
∞
X
νn+1 −1
n=1
S2∗ :=
,
! pq
n=1
λn
p
q
X
k=νn
S3∗ :=
,
∞
X
λn µ−1
n
n=1
k=1
∞
X
∞
X
λn µ−1
n
n=1
! pq
cqk
,
S4∗ :=
n
X
∞
X
n=1
p
! q−p
λk
,
p
! q−p
λk
,
k=n
λn
λn
µν n
p
q−p
,
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∗ < ∞.
242-08
2
L. L EINDLER
Recently, in [4], we proved that if
µn :=
Λ(1)
n
Cnp−q ,
where
∞
X
Cn :=
!1/q
cqk
and
Λ(1)
n
:=
k=n
then the sums S1 , S2 and S2∗ are already equiconvergent.
Furthermore if
!1/q
n
X
p−q
µn := Λ(2)
where C̃n :=
cqk
n C̃n ,
n
X
λk ,
k=1
and
k=1
Λ(2)
n
:=
∞
X
λk ,
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?
2. R ESULT
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, . . . ,
k=νm
then the sums S1 , S4 and S4∗ are equiconvergent.
3. L EMMA
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 P
numbers, and 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, . . . .
Then
(3.1)
σ1 :=
∞
X
an µ n < ∞
n=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp.
http://jipam.vu.edu.au/
E QUIVALENCE OF C OEFFICIENT C ONDITIONS
3
holds if and only if
∞
X
σ2 :=
(3.2)
κm Aανm < ∞.
m=1
Proof of Lemma 3.1. Before starting the proofs we note that the following inequality
m
X
(3.3)
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.
Furthermore we verify a useful inequality. If 0 ≤ a < b, 0 < α < 1 and
b α − aα
= α ξ α−1 ,
(3.4)
b−a
then
ξ ≥ α1/(1−α) b =: ξ0 ,
namely if a = 0 then ξ = ξ0 . Hence we get that
α ξ α−1 ≤ bα−1 .
(3.5)
Now we show that (3.1) implies (3.2). Since An & 0, thus, by (3.3),
∞
∞
∞
X
X
X
α
κm Aνm =
κm
(Aανn − Aανn+1 )
m=1
=
m=1
∞
X
n=m
(Aανn
−
Aανn+1 )
n=1
≤K
(3.6)
n
X
κm
m=1
∞
X
κn (Aανn − Aανn+1 ).
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 ≤
νn+1 −1
X
k=νn
!
ak
Aα−1
νn .
k=νn
This and (3.6) yield that
∞
X
κm Aανm
≤K
m=1
∞
X
νn+1 −1
κn Aα−1
νn
n=1
X
ak = K
∞ νn+1
X
X−1
ak µ k .
n=1 k=νn
k=νn
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
n=νm
κm Aανm ,
m=1
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp.
http://jipam.vu.edu.au/
4
L. L EINDLER
that is, (3.2) ⇒ (3.1) is verified.
Thus the proof is complete.
4. P ROOF OF T HEOREM 2.1
p
,
q
We shall use the result of Lemma 3.1 with α =
P∞ q
k=n ck and for νm ≤ n < νm+1
∞
X
µ n = µν m = λ m
(4.1)
an = cqn and κm = λm . Then An =
! p−q
q
cqk
.
k=νm
Then σ1 = S1 , thus by Lemma 3.1, S1 < ∞ implies that σ2 < ∞, that is,
! pq
! pq
νm+1 −1
∞
∞
∞
X
X
X
X
cqn
(4.2)
S4 =
λm
≤
λm
cqn
= σ2 .
m=1
n=νm
m=1
n=νm
Moreover, by (4.1),
S4∗ =
∞
X
n=1
λn

∞
 X

cqk
p
 q−p
! q−p
q 
=

k=νn
∞
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,
and (3.3), we obtain that
σ2 =
∞
X
λm Ap/q
νm ≤
m=1
=
νn+1 −1
n=1
k=νn
∞
X
X
! pq
cqk
n
X
n=1
νn+1 −1
n=m
k=νn
X
! pq
cqk
λm
m=1
! pq
νn+1 −1
λn
∞
X
λm
m=1
∞
X
≤K
∞
X
X
cqk
k=νn
= KS4 < ∞.
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
∞
X
log 2m
S4 =
< ∞,
m3
m=2
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp.
http://jipam.vu.edu.au/
E QUIVALENCE OF C OEFFICIENT C ONDITIONS
5
but S1 < ∞ and S4∗ < ∞ cannot be fulfilled simultaneously. Namely, then with a nondecreasing
sequence {µn } the conditions
∞
X
S1 =
m−6 µ2m < ∞
m=1
and
S4∗ =
∞
X
log2 m
<∞
µ
m
m=2
yield a trivial contradiction.
R EFERENCES
[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.
[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].
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp.
http://jipam.vu.edu.au/