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volume 7, issue 1, article 39, 2006.

Received 20 December, 2005;

accepted 21 January, 2006.

Communicated by:A.G. Babenko

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Journal of Inequalities in Pure and Applied Mathematics

A NEW EXTENSION OF MONOTONE SEQUENCES AND ITS APPLICATIONS

L. LEINDLER

Bolyai Institute University of Szeged Aradi vértanúk tere 1 H-6720 Szeged, Hungary.

EMail:leindler@math.u-szeged.hu

c

2000Victoria University ISSN (electronic): 1443-5756 371-05

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A New Extension of Monotone Sequences and its Applications

L. Leindler

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J. Ineq. Pure and Appl. Math. 7(1) Art. 39, 2006

http://jipam.vu.edu.au

Abstract

We define a new class of numerical sequences. This class is wider than any one of the classical or recently defined new classes of sequences of monotone type. Because of this generality we can generalize only the sufficient part of the classical Chaundy-Jolliffe theorem on the uniform convergence of sine series.

We also present two further theorems having conditions of sufficient type.

2000 Mathematics Subject Classification:26A15, 40-99, 40A05, 42A16.

Key words: Monotone sequences, Sequence ofγgroup bounded variation, Sine series.

This research was partially supported by the Hungarian National Foundation for Sci- entific Research under Grant Nos. T042462 and TS44782.

1. Introduction

In [3] we defined a subclass of the quasimonotone sequences(c_n≤K c_m, n≥ m), which is much larger than that of the monotone sequences and not compa- rable to the class of the classical quasimonotone sequences (see [6]). For this new class we have extended several results proved earlier only for monotone, quasimonotone or classical quasimonotone sequences. The definition of this class reads as follows: A null-sequence c (c_n → 0)belongs to the family of sequences of rest bounded variation (in brief,c∈RBV S)if

(1.1)

∞

X

n=m

|∆c_n| ≤K c_m (∆c_n =c_n−c_n+1)

holds for allm, whereK =K(c)is a constant depending only onc. Hereafter K will designate either an absolute constant or a constant depending on the indicated parameters, not necessarily the same at each occurrence.

Recently, in [7], we defined a new class of sequences as follows:

Letγ := {γ_n}be a positive sequence. A null-sequence cof real numbers satisfying the inequalities

(1.2)

∞

X

n=m

|∆c_n| ≤K γ_m

is said to be a sequence ofγ rest bounded variation(γ RBV S).

We emphasize that the classγ RBV S is no longer a subclass of the quasimonotone sequences. Namely, a sequencecsatisfying (1.2) may have infinitely many zero and negative terms, as well; but this is not the case ifcsatisfies (1.1).

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Very recently Le and Zhou [2] defined another new class of sequences using the following curious definition:

If there exists a natural numberN such that (1.3)

2m

X

n=m

|∆c_n| ≤K max

m≤n<m+N|c_n|

holds for all m, then c belongs to the class GBV S, in other words, c is a sequence of group bounded variation.

The classGBV S is an ingenious generalization of RBV S, moreover it is wider than the class of the classical quasimonotone sequences c_n+1 ≤c_n 1 + ^α_n

, too.

In [2], among others, they verified that the monotonicity condition in the classical theorem of Chaundy and Jolliffe [1] can be replaced by their condition (1.3). Herewith they improved our result, namely that in [5], we verified this by condition (1.1).

The aim of the present work is to unify the advantages of the definitions (1.2) and (1.3). We define a further new class of sequences, to be denoted by γ GBV S, which is wider than any one of the classesGBV Sandγ RBV S.

A null-sequencecbelongs toγ GBV S if (1.4)

2m

X

n=m

|∆cn| ≤K γm, m= 1,2, . . .

holds, whereγis a given sequence of nonnegative numbers.

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We underline that the sequenceγ satisfying (1.4) may have infinitely many zero terms, too; but not in (1.2). We also emphasize that the condition (1.4) gives the greatest freedom for the terms of the sequencescandγ.

As a first application we shall give a sufficient condition for the uniform convergence of the series

(1.5)

∞

X

n=1

b_n sinnx,

whereb:={b_n}belongs to a certain class ofγ GBV S.

Utilizing the benefits of the sequences of γ GBV S we present two further generalizations of theorems proved earlier for sequences ofγ RBV S.

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2. Theorems

We verify the following theorems:

Theorem 2.1. Letγ :={γn}be a sequence of nonnegative numbers satisfying the condition γ_n = o(n⁻¹). If a sequence b := {b_n} ∈ γ GBV S, then the series (1.5) is uniformly convergent, and consequently its sum functionf(x)is continuous.

Compare Theorem2.1to the mentioned theorem of Chaundy and Jolliffe and two theorems of ours [5, Theorem A and Theorem 1] and [8, Theorem 1]. The cited theorems proved their statements for monotone sequences, b ∈ RBV S andb∈γ RBV S, respectively.

Remark 1. It is easy to see that if b_n = n⁻¹ and γ_n = n⁻¹, then {b_n} ∈ γ GBV S and the series (1.5) does not converge uniformly. This shows that the assumptionγ_n=o(n⁻¹)cannot be weakened generally.

Theorem 2.2. Let β :={η_n}be a sequence of nonnegative numbers satisfying the condition η_n = O(n⁻¹). If a sequence b := {b_n} ∈ β RBV S, then the partial sums of the series (1.5) are uniformly bounded.

We note that for a monotone null-sequenceb, moreover forb∈RBV Sand b ∈ γ RBV S, the assertion of Theorem2.2 can be found in [10, Chapter V,

§1], in [5, Theorem 2] and [8, Theorem 2].

Before formulating Theorem 2.3 we recall the following definition. A se- quenceβ :={β_n}of positive numbers is called quasi geometrically increasing (decreasing) if there exist natural numbersµandK =K(β) ≥1such that for

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all natural numbersn

β_n+µ≥2β_nandβ_n ≤K β_n+1

β_n+µ ≤ 1

2β_nandβ_n+1 ≤K β_n

. Theorem 2.3. Ifc :={c_n} ∈ β GBV S, or belongs toγ GBV S, whereβ and γ have the same meaning as in Theorems2.1and2.2, furthermore the sequence {n_m}is quasi geometrically increasing, then the estimates

(2.1)

∞

X

j=1

nj+1−1

X

k=nj

c_k sinkx

≤K(c,{n_m}),

or (2.2)

∞

X

j=m

nj+1−1

X

k=nj

c_k sinkx

=o(1), m→ ∞,

hold uniformly inx, respectively.

The root of (2.1) goes back to Telyakovski˘ı [9, Theorem 2] and two generalizations of it can be found in [5] and [8].

We note that, in general, (2.1) does not imply (2.2), see the Remark in [8].

It is clear that the “smallest” classγ GBV Swhich includes a given sequence c:={c_n}is the one, where

γn :=

2n

X

k=n

|∆ck|, n= 1,2, . . .

In regard to this, it is plain, that our theorems convey the following consequence.

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Corollary 2.4. The assertions of our theorems for an individual sequence b hold true under the assumptions

(2.3)

2n

X

k=n

|∆bk|=o(n⁻¹)

and 2n

X

k=n

|∆b_k|=O(n⁻¹),

respectively.

However, in my view, our theorems give a better perspicuity than Corollary 2.4 does; the arrangement of the proofs are more convenient with our method, furthermore the assumptions of Corollary 2.4give conditions only for an individual sequence, and not for a class of sequences.

We also remark that e.g. the condition (2.3) is not a necessary one for uniform convergence. See the series

∞

X

n=1

2⁻ⁿsin 2ⁿx.

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3. Lemmas

Lemma 3.1 ([4]). For any positive sequence{β_n}the inequalities

∞

X

n=m

β_n ≤K β_m, m = 1,2, . . .; K ≥1,

hold if and only if the sequence{βn}is quasi geometrically decreasing.

Lemma 3.2. Letρ:={ρ_n}be a nonnegative sequence withρ_n =O(n⁻¹), and letδ :={δ_n}belong toρ GBV S. If a complex seriesP∞

n=1a_nsatisfies the Abel condition, i.e., if there exists a constantAsuch that for allm ≥1,

m

X

n=1

a_n

≤A,

then for anyµ≥m,

(3.1)

µ

X

n=m

a_nδ_n

≤6K(ρ)A ε_mm⁻¹,

where K(ρ)denotes the constant appearing in the definition of ρ GBV S, fur- thermore

ε_n:= sup

k≥n

k ρ_k. Consequently, ifε_m =o(m), then the seriesP∞

n=1a_nδ_nconverges.

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Proof. First we show that

(3.2) |δ_m| ≤

∞

X

n=m

|∆δ_n| ≤2K(ρ)ε_mm⁻¹.

Sinceδ_ntends to zero, the first inequality in (3.2) is obvious; and becausen ρ_n is bounded, thusδ∈ρ GBV S implies that

∞

X

n=m

|∆δ_n| ≤

∞

X

`=0 2^`+1m

X

n=2^`m

|∆δ_n| ≤

∞

X

`=0

K(ρ)ρ₂^`_m

≤K(ρ)

∞

X

`=0

ε_m(2^`m)⁻¹ = 2K(ρ)m⁻¹ε_m, (3.3)

and this proves (3.2).

Next we verify (3.1). Using the notation α_n:=

n

X

k=1

a_k, (3.2) and the assumptions of Lemma3.2, we get that

µ

X

n=m

a_nδ_n

=

µ−1

X

n=m

α_n(δ_n−δ_n+1) +α_µδ_µ−αm−1δ_m

≤A

µ−1

X

n=m

|∆δ_n|+|δ_µ|+|δ_m|

!

≤6AK(ρ)ε_mm⁻¹,

which proves (3.1).

The proof is complete.

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4. Proofs

Proof of Theorem2.1. Denote ε_n := sup

k≥n

k γ_k and r_n(x) :=

∞

X

k=n

b_k sinkx.

In view of the assumptionγ_m =o(m⁻¹)we have thatε_n →0asn → ∞. Thus it is sufficient to verify that

(4.1) |r_n(x)| ≤K ε_n

holds for alln.

Sincer_n(kπ) = 0it suffices to prove (4.1) for0< x < π.

LetN be the integer for which

(4.2) π

N + 1 < x≤ π N. First we show that ifk ≥nthen

(4.3) k|b_k| ≤K ε_n, n = 1,2, . . .

Sinceb_mandm γ_mtend to zero, thus the assumptionb ∈γ GBV S implies that

|b_k| ≤

2k−1

X

i=k

|∆b_i|+|b_2k| ≤

1

X

`=0

2^`+1k−1

X

i=2^`k

|∆b_i|+|b_4k| ≤ · · ·

≤K

∞

X

`=0

γ₂^`_k =:σk. (4.4)

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By the definition ofε_nandk ≥nwe have that

2^`k γ₂^`_k≤ε_n, `= 1,2, . . . , thus it is clear that

σ_k≤2Kε_n/k;

this and (4.4) proves (4.3).

Now we turn back to the proof of (4.1). Let r_n(x) =

n+N−1

X

k=n

+

∞

X

k=n+N

!

b_ksinkx=:r⁽¹⁾_n (x) +r_n⁽²⁾(x).

Then, by (4.2) and (4.3), (4.5) |r_n⁽¹⁾(x)| ≤x

n+N−1

X

k=n

k|b_k| ≤K x N ε_n≤K π ε_n.

A similar consideration as in (3.3) gives that for anym ≥n

∞

X

k=m

|∆b_k| ≤K ε_n/m.

Using this, (4.2), (4.3) and the well-known inequality D_n(x) :=

n

X

k=1

sinkx

≤ π x,

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furthermore summing by parts, we get that

|r_n⁽²⁾(x)| ≤

∞

X

k=n+N

|∆bk|Dk(x) +|bn+N|Dn+N−1(x)

≤2K ε_n n+N

π

x ≤2K εn. (4.6)

The inequalities (4.5) and (4.6) imply (4.1), that is, the series (1.5) is uniformly convergent.

Proof of Theorem2.2. In the proof of Theorems 2.2 and 2.3 we shall use the notations of the proof of Theorem2.1. The conditionη_n =O(n⁻¹)implies that the sequence{ε_n}is bounded, i.e.ε_n≤ K.This, (4.2) and (4.3) imply that for anym≤N

m

X

k=1

b_ksinkx

≤

N

X

k=1

|b_k|kx≤K x N ≤K π,

furthermore, ifm > N then, by (4.1),

m

X

k=N+1

b_ksinkx

≤ |r_N+1(x)|+|r_m+1(x)| ≤2K ε₁.

The last two estimates clearly prove Theorem2.2.

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Proof of Theorem2.3. First we verify (2.1). Let us suppose that

(4.7) n_i ≤N < n_i+1.

Sincec ∈ β GBV S andη_n = O(n⁻¹),we get, as in the proof of Theorem2.2 withc_nin place ofb_n, that

(4.8)

i−1

X

j=1

nj+1−1

X

k=nj

c_ksinkx

+

N

X

k=ni

c_ksinkx

≤

i−1

X

j=1 nj+1−1

X

k=nj

|c_k|kx+

N

X

k=ni

|c_k|kx≤Kπ.

Next applying Lemma3.2withρ=β, δ_n=c_nanda_n = sinnx, we get that

σ^∗_N :=

ni+1−1

X

k=N+1

c_ksinkx

+

∞

X

j=i+1

nj+1−1

X

k=nj

c_ksinkx

≤K (

εN+1(N + 1)⁻¹x⁻¹+x⁻¹

∞

X

j=i+1

εnjn⁻¹_j )

≤K (

ε_N +N ε_N

∞

X

j=i+1

n⁻¹_j )

≤K ε_N (

1 +N

∞

X

j=i+1

n⁻¹_j )

. (4.9)

Since the sequence {n_j} is quasi geometrically increasing, so {n⁻¹_j } is quasi

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geometrically decreasing, therefore, Lemma3.1and (4.7) imply that (4.10)

∞

X

j=i+1

n⁻¹_j ≤K N⁻¹,

whence, by (4.9) andη_n =O(n⁻¹),

(4.11) σ_N^∗ ≤K ε_N <∞

follows. Herewith (2.1) is proved.

Ifc∈ γ GBV Sthen, byγ_n =o(n⁻¹), ε_n→0, thus, withmin place ofN, (4.9), (4.10) and (4.11) immediately verify (2.2).

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References

[1] T.W. CHAUNDY AND A.E. JOLLIFFE, The uniform convergence of a certain class of trigonometric series, Proc. London Math. Soc., 15 (1916), 214–216.

[2] R.J. LE AND S.P. ZHOU, A new condition for uniform convergence of certain trigonometric series, Acta Math. Hungar., 10(1-2) (2005), 161–

169.

[3] L. LEINDLER, Embedding results pertaining to strong approximation of Fourier series. II, Analysis Math., 23 (1997), 223–240.

[4] L. LEINDLER, On the utility of power-monotone sequences, Publ. Math.

Debrecen, 55(1-2) (1999), 169–176.

[5] L. LEINDLER, On the uniform convergence and boundedness of a certain class of sine series, Analysis Math., 27 (2001), 279–285.

[6] L. LEINDLER, A new class of numerical sequences and its applications to sine and cosine series, Analysis Math., 28 (2002), 279–286.

[7] L. LEINDLER, Embedding results regarding strong approximation of sine series, Acta Sci. Math. (Szeged), 71 (2005), 91–103.

[8] L. LEINDLER, A note on the uniform convergence and boundedness of a new class of sine series, Analysis Math., 31 (2005), 269–275.

[9] S.A. TELYAKOVSKI˘I, On partial sums of Fourier series of functions of bounded variation, Proc. Steklov Inst. Math., 219 (1997), 372–381.

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[10] A. ZYGMUND, Trigonometric Series. Vol. I, University Press (Cam- bridge, 1959).