(1)ON EQUIVALENCE OF COEFFICIENT CONDITIONS

ON EQUIVALENCE OF COEFFICIENT CONDITIONS. II

L. LEINDLER BOLYAIINSTITUTE

UNIVERSITY OFSZEGED

ARADI VÉRTANÚK TERE1 H-6720 SZEGED, HUNGARY

leindler@math.u-szeged.hu

Received 28 August, 2008; accepted 16 September, 2008 Communicated by H. Bor

ABSTRACT. 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. INTRODUCTION

In the papers [2], [3] and [4] we have studied the relations of the following sums:

S₁ :=

∞

X

n=1

c^q_nµ_n,

S₂ :=

∞

X

n=1

λ_n

∞

X

k=n

c^q_k

!^p_q

, S₂^∗ :=

∞

X

n=1

λ_n µ⁻¹_n

n

X

k=1

λ_k

!_q−p^p ,

S₃ :=

∞

X

n=1

λ_n

n

X

k=1

c^q_k

!^p_q

, S₃^∗ :=

∞

X

n=1

λ_n µ⁻¹_n

∞

X

k=n

λ_k

!_q−p^p ,

S₄ :=

∞

X

n=1

λ_n

νn+1−1

X

k=νn

c^q_k

!

p q

, S₄^∗ :=

∞

X

n=1

λ_n λ_n

µνn

_q−p^p ,

where0< p < q, λ:={λ_n}andc:={c_n}are sequences of nonnegative numbers,ν :={ν_m} is a subsequence of natural numbers, and µ := {µ_n} is a certain nondecreasing sequence of positive numbers.

In [2] we verified thatS₂ <∞if and only if there exists aµsatisfying the conditionsS₁ <∞ andS₂^∗ <∞.SimilarlyS₃ <∞if and only ifS₁ <∞andS₃^∗ <∞.

In [3] we showed thatS₄ <∞if and only if there exists aµsuch thatS₁ <∞andS₄^∗ <∞.

242-08

Recently, in [4], we proved that if

µ_n := Λ⁽¹⁾_n C_n^p−q, where C_n :=

∞

X

k=n

c^q_k

!1/q

and Λ⁽¹⁾_n :=

n

X

k=1

λ_k, then the sumsS₁, S₂andS₂^∗ are already equiconvergent.

Furthermore if

µ_n := Λ⁽²⁾_n C˜_n^p−q, where C˜_n :=

n

X

k=1

c^q_k

!1/q

and Λ⁽²⁾_n :=

∞

X

k=n

λ_k, then the sumsS₁, S₃andS₃^∗ 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{µ_n}are determined, herewith they state more than the outcomes of [2] and [3], where only the existence of a sequence{µ_n}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 sumsS₂ andS₃, but not forS4. 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 par- ticular,λshould be quasi geometrically increasing, that is, we assume that there exist a natural numberN andK ≥1such thatλ_n+N ≥2λ_nandλ_n≤Kλ_n+1 hold for alln.

Then we can give an explicit sequenceµsuch that the sumsS₁, S₄andS₄^∗are already equicon- vergent. We also show that without some additional requirement onλthe equiconvergence does not hold. See the last part. Thus the following open problem can be raised: What is the weakest additional assumption on sequenceλwhich ensures the equiconvergence of these sums?

2. RESULT

Theorem 2.1. If0 < p < q, c :={c_n}is a sequence of nonnegative numbers,ν :={ν_m}is a subsequence of natural numbers, andλ :={λ_n}is a quasi geometrically increasing sequence, and forν_m ≤n < ν_m+1

µ_n :=λ_m

∞

X

k=νm

c^q_k

!^p_q−1

, m= 0,1, . . . ,

then the sumsS₁, S₄andS₄^∗ are equiconvergent.

3. LEMMA

In order to verify our theorem, first we shall prove a lemma regarding the equiconvergence of two special series.

Lemma 3.1. Let0< α <1, a:={a_n}be a sequence of nonnegative numbers,ν :={ν_m}be a subsequence of natural numbers, andκ:={κ_m}be a quasi geometrically increasing sequence.

Furthermore letA_k :=P∞

n=k a_n,and forν_m ≤n < ν_m+1let µ_n:=κ_mA^α−1_νm , m= 0,1, . . . . Then

(3.1) σ₁ :=

∞

X

n=1

a_nµ_n<∞

holds if and only if

(3.2) σ₂ :=

∞

X

m=1

κ_mA^α_νm <∞.

Proof of Lemma 3.1. Before starting the proofs we note that the following inequality (3.3)

m

X

n=1

κ_n≤K κ_m

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. If0≤a < b, 0< α <1and

(3.4) b^α−a^α

b−a =α ξ^α−1, then

ξ≥α^1/(1−α)b =:ξ₀, namely ifa= 0thenξ =ξ₀.Hence we get that

(3.5) α ξ^α−1 ≤b^α−1.

Now we show that (3.1) implies (3.2). SinceA_n&0,thus, by (3.3),

∞

X

m=1

κ_mA^α_ν

m =

∞

X

m=1

κ_m

∞

X

n=m

(A^α_ν

n−A^α_ν

n+1)

=

∞

X

n=1

(A^α_νn−A^α_νn+1)

n

X

m=1

κ_m

≤K

∞

X

n=1

κ_n(A^α_ν

n−A^α_ν

n+1).

(3.6)

Using the relations (3.4) and (3.5) we obtain that A^α_νn −A^α_νn+1 =

νn+1−1

X

k=νn

a_k

!

α ξ^α−1 ≤

νn+1−1

X

k=νn

a_k

! A^α−1_νn . This and (3.6) yield that

∞

X

m=1

κ_mA^α_νm ≤K

∞

X

n=1

κ_nA^α−1_νn

νn+1−1

X

k=νn

a_k =K

∞

X

n=1 νn+1−1

X

k=νn

a_kµ_k. Herewith the implication (3.1)⇒(3.2) is proved.

The proof of (3.2)⇒(3.1) is very easy. Namely

∞

X

n=ν1

a_nµ_n =

∞

X

m=1 νm+1−1

X

n=νm

a_nµ_n

=

∞

X

m=1

κ_mA^α−1_ν

m

νm+1−1

X

n=νm

a_n

≤

∞

X

m=1

κ_mA^α_ν

m,

that is, (3.2)⇒(3.1) is verified.

Thus the proof is complete.

4. PROOF OFTHEOREM2.1

We shall use the result of Lemma 3.1 with α = ^p_q, a_n = c^q_n and κ_m = λ_m. Then A_n = P∞

k=nc^q_kand forν_m ≤n < ν_m+1

(4.1) µ_n=µ_νm =λ_m

∞

X

k=νm

c^q_k

!^p−q_q .

Thenσ₁ =S₁, thus by Lemma 3.1,S₁ <∞implies thatσ₂ <∞, that is,

(4.2) S₄ =

∞

X

m=1

λ_m

νm+1−1

X

n=νm

c^q_n

!

p q

≤

∞

X

m=1

λ_m

∞

X

n=νm

c^q_n

!^p_q

=σ₂.

Moreover, by (4.1),

S₄^∗ =

∞

X

n=1

λn







∞

X

k=νn

c^q_k

!^q−p_q 





p q−p

=

∞

X

n=1

λn

∞

X

k=νn

c^q_k

!^p_q

=σ2, thusS₁ <∞implies that bothS₄ <∞andS₄^∗ <∞hold.

Conversely, ifS₄ <∞,then it suffices to show thatσ₂ =S₄^∗ <∞also holds.

Applying the inequality

X a_kα

≤X

a^α_k, 0< α≤1, a_k ≥0, and (3.3), we obtain that

σ₂ =

∞

X

m=1

λ_mA^p/q_νm ≤

∞

X

m=1

λ_m

∞

X

n=m

νn+1−1

X

k=νn

c^q_k

!

p q

=

∞

X

n=1

νn+1−1

X

k=νn

c^q_k

!

p q n

X

m=1

λ_m

≤K

∞

X

n=1

λ_n

νn+1−1

X

k=νn

c^q_k

!

p q

=KS4 <∞.

This, (4.2) and, by Lemma 3.1, the implicationσ₂ <∞ ⇒σ₁ =S₁ <∞complete the proof of Theorem 2.1.

Proof of the necessity of some additional assumption onλ. Letp= 1, q = 2, λ_n= logn, ν_n= nand

c_n:= m⁻³ if n= 2^m, 0 otherwise.

Then

S₄ =

∞

X

m=2

log 2^m m³ <∞,

butS₁ <∞andS₄^∗ <∞cannot be fulfilled simultaneously. Namely, then with a nondecreasing sequence{µ_n}the conditions

S1 =

∞

X

m=1

m⁻⁶µ2^m <∞ and

S₄^∗ =

∞

X

m=2

log²m µm

<∞

yield a trivial contradiction.

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. (Za- greb), 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].