ON EQUIVALENCE OF COEFFICIENT CONDITIONS. II

(1)

Equivalence of Coefficient Conditions

L. Leindler vol. 9, iss. 3, art. 83, 2008

Title Page

Contents

JJ II

J I

Page1of 12 Go Back Full Screen

Close

ON EQUIVALENCE OF COEFFICIENT CONDITIONS. II

L. LEINDLER

Bolyai Institute, University of Szeged Aradi vértanúk tere 1,

H-6720 Szeged, Hungary

EMail:leindler@math.u-szeged.hu

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.

(2)

Title Page Contents

JJ II

J I

Close

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

µ_ν_n _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 that S₂ < ∞ if and only if there exists a µ satisfying the conditionsS₁ < ∞ andS₂^∗ < ∞. SimilarlyS₃ < ∞if and only if S₁ < ∞ and S₃^∗ <∞.

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

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,

(4)

Title Page Contents

JJ II

J I

Close

then the sumsS₁, S₂ andS₂^∗ are already equiconvergent.

Furthermore if

µn := 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

λk, then the sumsS₁, S₃ andS₃^∗ 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 sums S2andS3, 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 particular, λ should be quasi geometrically increasing, that is, we assume that there exist a natural numberN andK ≥ 1such thatλ_n+N ≥ 2λ_n and λ_n≤Kλ_n+1hold for alln.

Then we can give an explicit sequence µ such that the sumsS₁, 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 sequenceλ which ensures the equiconvergence of these sums?

(5)

Title Page Contents

JJ II

J I

Close

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, S₄ andS₄^∗ are equiconvergent.

(6)

Title Page Contents

JJ II

J I

Close

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 :=

{νm}be a subsequence of natural numbers, andκ:={κm}be a quasi geometrically increasing sequence. Furthermore letAk :=P∞

n=k an,and forνm ≤n < νm+1let µ_n:=κ_mA^α−1_ν

m , 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_ν_m <∞.

Proof of Lemma3.1. 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.

(7)

Title Page Contents

JJ II

J I

Close

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). SinceAn &0,thus, by (3.3),

∞

X

m=1

κmA^α_ν_m =

∞

X

m=1

κm

∞

X

n=m

(A^α_ν_n−A^α_ν_n+1)

=

∞

X

n=1

(A^α_ν_n−A^α_ν

n+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_ν

n −A^α_ν

n+1 =

νn+1−1

X

k=νn

a_k

!

α ξ^α−1 ≤

νn+1−1

X

k=νn

a_k

! A^α−1_ν

n . This and (3.6) yield that

∞

X

m=1

κ_mA^α_ν_m ≤K

∞

X

n=1

κ_nA^α−1_ν_n

νn+1−1

X

k=νn

a_k =K

∞

X

n=1 νn+1−1

X

k=νn

a_kµ_k.

(8)

Title Page Contents

JJ II

J I

Close

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.

(9)

Title Page Contents

JJ II

J I

Close

4. Proof of Theorem 2.1

We shall use the result of Lemma 3.1 withα = ^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_ν_m =λ_m

∞

X

k=νm

c^q_k

!^p−q_q .

Thenσ₁ =S₁, thus by Lemma3.1,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

λ_n







∞

X

k=νn

c^q_k

!^q−p_q 





p q−p

=

∞

X

n=1

λ_n

∞

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,

(10)

Title Page Contents

JJ II

J I

Close

and (3.3), we obtain that σ₂ =

∞

X

m=1

λ_mA^p/q_ν_m ≤

∞

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

λn

νn+1−1

X

k=νn

c^q_k

!

p q

=KS4 <∞.

This, (4.2) and, by Lemma 3.1, the implication σ₂ < ∞ ⇒ σ₁ = S₁ < ∞ complete the proof of Theorem2.1.

Proof of the necessity of some additional assumption onλ. Letp= 1, q = 2, λn = logn, νn=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 <∞

(11)

Title Page Contents

JJ II

J I

Close

and

S₄^∗ =

∞

X

m=2

log²m µ_m <∞ yield a trivial contradiction.

(12)

Title Page Contents

JJ II

J I

Close

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.

[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].