International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 1 (2014), 36-44
http://www.etamaths.com

ON THE BEHAVIOR NEAR THE ORIGIN OF A SINE SERIES

WITH COEFFICIENTS OF MONOTONE TYPE

XHEVAT Z. KRASNIQI

Abstract. In this paper we have obtained some asymptotic equalities of the

sum function of a trigonometric sine series expressed in terms of its special

type of coefficients.

1. Introduction

Let us consider the sine series

(1.1)

∞∑
m=1

am sin mx

with coefficients tending to zero and such that the sequence {am} satifsies condition
4am = am−am+1 ≥ 0 or 42am = 4am−4am+1 ≥ 0 for all m. It is a well-known
fact that under such conditions the series (1.1) converges for all x (see [12], page
95). We denote by g(x) its sum.

As usually we write g(u) ∼ h(u),u → 0 if there exist absolute positive constants
A and B such that Ah(u) ≤ g(u) ≤ Bh(u) is in a neighborhood of the point u = 0,
and write g(u) ≈ h(u) if limu→0

g(u)
h(u)

= 1. Likewise, throughout this paper the

constants in the O-expression denote positive absolute constants and they may be
different in different relations.

Several authors have investigated the behavior of the sum g(x) near the origin
expressed in terms of the coefficients am. Seemingly, the first was Young [11] who
consider this problem, and he was concerned solely about estimates of |g(x)| from
above. Then Salem ([3], [4], Theorem 1) proved that if the sequence {mam} is
monotone decreasing, then the following order equality holds

g(x) ∼
∑̀
m=1

mamx,

where x ∈ I` :=
(

π
`+1

, π
`

]
,` = 1, 2, . . . , x → 0.

Later on, Aljančić, Bojanić and Tomić ([5], Theorem 2) give asymptotic expres-
sion for g(x) as x → 0, when the coefficients am are convex (42am ≥ 0) and can
be represent as the values A(m) of a slowly varying (in Karamata’s sense) function

2010 Mathematics Subject Classification. 42A20, 42A32.
Key words and phrases. Sine series, (k, s)-monotone, convex sequence, asymptotic equality.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

36



ON THE BEHAVIOR NEAR THE ORIGIN OF A SINE SERIES 37

A(z), i.e. for each t > 0

(1.2) lim
z→∞

A(tz)

A(z)
= 1.

Their result is equivalent to the following statement which can be deduce from
one result given by Telyakovskĭı ([6], Theorem 2) and it is formulated as a corollary
in this form:

Corollary 1.1. Suppose that the coefficients am of the series (1.1) are convex and
that am = A(m), for a slowly varying function A(z). Then the following asymptotic
equality holds true:

g(x) ≈ a`
1

x
, x ∈ I`, x → 0.

Telyakovskĭı deduced this result after the proof, in the same paper, of the fol-
lowing two theorems:

Theorem 1.1. Assume that am ↓ 0. Then for x ∈ I` the following estimate is
valid

g(x) =
∑̀
m=1

mamx + O

(
1

`3

∑̀
m=1

m3am

)
.

Theorem 1.2. Let am → 0 and let the sequence {am} be convex. If x ∈ I`, where
` ≥ 11, then the following estimate holds true

a`
2

cot
x

2
+

1

2`

`−1∑
m=1

m24am ≤ g(x) ≤
a`
2

cot
x

2
+

6

`

`−1∑
m=1

m24am.

Note also that the above theorems as well as some of [1] are generalized and
extended in [7]-[10].

For an integer k ≥ 0 and a real sequence {am}∞m=0 denote

4kam =
k∑
i=0

(−1)iCikam+i (40am = am),

{4}k am =
k∑
i=0

Cikam+i ({4}0 am = am).

Definition 1.1 ([2]). A sequence {am}∞m=0 is said to be (k,s)-monotone if am → 0
as m →∞ and 4k ({4}s am) ≥ 0, for some k ≥ 0,s ≥ 0 and all m.

It is easy to see that that if a sequence {am} (am → 0 as m → ∞) is non-
increasing, then it is (1,s)-monotone for all s = 0, 1, 2, . . . . The converse statement
is not always true. For example, if we consider the sequence {am} such that am → 0
as m →∞ and a2m = 0, a2m+1 ≥ a2m+3 for m = 0, 1, 2, . . . , then this sequence is
not non-increasing but it is (1, 1)-monotone.

Chronologically this definition arises the following question: What is the behav-
ior near the origin of the series (1.1) with (k,s)-monotone coefficients? The answer
to this question is the main goal of this paper. Precisely, we shall answer to this
question only considering the cases when the series (1.1) has: (1, 1)-monotone, or
(1, 2)-monotone, or (2, 1)-monotone, or (2, 2)-monotone coefficients.

For the proof of our results we need the following two lemmas proved in [2].



38 XHEVAT Z. KRASNIQI

Lemma 1.1. Let {am}∞m=0 be a sequence such that am → 0 as m → ∞ and
4kam ≥ 0 for some k ≥ 1 and all m. Then for all r = 0, 1, . . . ,k−1 and all m the
following inequality 4ram ≥ 0 holds.

Lemma 1.2. Let {am}∞m=0 be a (k,s)-monotone sequence. If k = 1,s = 1 or s = 2,
then

g(x) =
a0
2

(
1 − tan

x

2

)
+

1(
2 cos x

2

)s ∞∑
m=1

{4}sam−1 sin (ms− 2 + s)
x

2
,

allmost everywhere.

Lemma 1.3. Let Bm(x) =
∑m
i=0 sin (i− 1)

x
2

. Then the following estimates hold:∣∣Bm(x)∣∣ ≤ 2π
x
, 0 < x ≤ π.

Proof. After some elementary calculation we have∣∣Bm(x)∣∣ = ∣∣∣∣ 12 sin x
2

m∑
i=0

[
cos (i− 2)

x

2
− cos

ix

2

]∣∣∣∣
=

∣∣∣∣cos x2 + cos x− cos (m− 1) x2 − cos mx22 sin x
2

∣∣∣∣
≤

2∣∣ sin x
2

∣∣ ≤ 2πx , 0 < x ≤ π.
�

2. Main Results

The following theorem considers sine series with (1, 1)-monotone sequence.

Theorem 2.1. Assume that {am}∞m=1 is a (1, 1)-monotone sequence. Then for
x ∈ I` the following estimate is valid

g(x) =
1

2 cos x
2

{
1

2

∑̀
m=1

m{4}1amx + O

(
1

`3

∑̀
m=1

m3{4}1am

)}
.(2.1)

Proof. By the Lemma 1.2 (a0 = 0) we have

g(x) =
1

2 cos x
2

∞∑
m=1

{4}1am−1 sin (m− 1)
x

2
.(2.2)

Then the use of Abel’s transformation gives

H(x) = lim
p→∞

{
p−1∑
m=1

4({4}1am−1) Bm(x) + {4}1ap−1Bp(x) + {4}1a0 sin
x

2

}

=

∞∑
m=1

4({4}1am−1) Bm(x) + {4}1a0 sin
x

2
:= H

(1)
` (x) + H

(2)
` (x),(2.3)

where

H
(1)
` (x) =

`+1∑
m=1

4({4}1am−1) Bm(x) + {4}1a0 sin
x

2
,



ON THE BEHAVIOR NEAR THE ORIGIN OF A SINE SERIES 39

and

H
(2)
` (x) =

∞∑
m=`+2

4({4}1am−1) Bm(x).

Let us estimate first H
(1)
` (x). Based on Lemma 1.3, our assumption 4({4}1am) ≥

0 for all m, the well-known relation sin t = t + O(t3), as t → 0, and x ∈ I` we have

H
(1)
` (x) =

`+1∑
m=1

({4}1am−1 −{4}1am) Bm(x) + {4}1a0 sin
x

2

=
∑̀
m=0

{4}1am
[
Bm+1(x) −Bm(x)

]
−{4}1a`+1B`+1(x)

=
∑̀
m=1

{4}1am sin
mx

2
+

2π

x
{4}1a`+1

=
1

2

∑̀
m=1

m{4}1amx + O

(
1

`3

∑̀
m=1

m3{4}1am

)
+ O (`{4}1a`) .

By virtue of monotonicity of {4}1am we obtain

`{4}1a` ≤
4

`3

{
`(` + 1)

2

}2
{4}1a` ≤

4

`3

∑̀
m=1

m3{4}1am.

Thus,

H
(1)
` (x) =

1

2

∑̀
m=1

m{4}1amx + O

(
1

`3

∑̀
m=1

m3{4}1am

)
.(2.4)

Furthermore, since x ∈ I` and |Bm(x)| = O
(
1
x

)
by the Lemma 1.2, we notice

that

H
(2)
` (x) = O

(
1

x

∞∑
m=`+2

({4}1am−1 −{4}1am)

)
= O ((` + 1){4}1a`+1) = O (`{4}1a`)

= O

(
1

`3

∑̀
m=1

m3{4}1am

)
.(2.5)

Finally, relations (2.2)-(2.5) prove completely estimation (2.1). �

Corollary 2.1. Let {am}∞m=1 be a (1, 1)-monotone sequence and the series
∞∑
m=1

m (am + am+1)

converges. Then the following asymptotic equality

lim
x→0

g(x)

x
=

1

4

∞∑
m=1

m (am + am+1)

holds true.



40 XHEVAT Z. KRASNIQI

Proof. In accordance with Theorem 2.1 it is enough to prove that

1

`2

∑̀
m=1

m3{4}1am → 0, as ` →∞.

Indeed, for an arbitrary natural number M we can write

1

`2

∑̀
m=1

m3{4}1am ≤
1

`2

M∑
m=1

m3{4}1am +
∞∑

m=M+1

m{4}1am.

If a number ε > 0 be chosen, then by hypotesis a number M = M(ε) exists, such
that

∞∑
m=M+1

m{4}1am <
ε

2
.

Likewise, for all sufficiently large `

1

`2

M∑
m=1

m3{4}1am <
ε

2
.

Then obviously, for such ` we have

1

`2

∑̀
m=1

m3{4}1am <
ε

2
+
ε

2
= ε.

�

The following statements can be proved similarly therefore we will skip their
proofs.

Theorem 2.2. Assume that {am}∞m=1 is a (1, 2)-monotone sequence. Then for
x ∈ I` the following estimate is valid

g(x) =
1(

2 cos x
2

)2
{∑̀
m=0

(m + 1){4}2amx + O

(
1

`3

∑̀
m=0

(m + 1)3{4}2am

)}
.

Corollary 2.2. Suppose that {am}∞m=1 is a (1, 2)-monotone sequence and the series
∞∑
m=0

(m + 1) (am + 2am+1 + am+2)

converges. Then the following asymptotic equality

lim
x→0

g(x)

x
=

1

4

∞∑
m=0

(m + 1) (am + 2am+1 + am+2)

holds true.

The proof of the next statement is more complicated and that is why we will
sketch it in more details.



ON THE BEHAVIOR NEAR THE ORIGIN OF A SINE SERIES 41

Theorem 2.3. Assume that {am}∞m=1 is a (2, 2)-monotone sequence. Then for
x ∈ I`, ` ≥ 11 the following estimate is valid

{4}2a`−1
2

cot
x

2
+

1

2`

`−1∑
m=1

m24({4}2am−1)

≤ g(x)
(

2 cos
x

2

)2
≤
{4}2a`−1

2
cot

x

2
+

6

`

`−1∑
m=1

m24({4}2am−1) .

Proof. By the Lemma 1.1 the condition 42 ({4}2am) ≥ 0 implies 4({4}2am) ≥ 0.
Therefore by the Lemma 1.2 we have

g(x) =
1(

2 cos x
2

)2 ∞∑
m=1

{4}2am−1 sin mx.

Applying Abel’s transformation we obtain

(2.6) g(x) =
1(

2 cos x
2

)2 ∞∑
m=1

4({4}2am−1) D̃m(x),

where D̃m(x) =
∑m
i=1 sin ix is the conjugate Dirichlet kernel.

For x ∈ (0,π] and m = 0, 1, 2, . . . , introduce the functions

ϕm(x) := −
cos (m + 1/2) x

2 sin x/2

and

ψm(x) :=

m∑
i=0

ϕi(x) = −
sin (m + 1) x

4 sin2(x/2)
.

Denoting H(x) :=
∑∞
m=1 4({4}2am−1) D̃m(x) one can write

H(x) =

`−1∑
m=1

4({4}2am−1) D̃m(x)

+

∞∑
m=`

4({4}2am−1)
(

1

2
cot

x

2
+ ϕm(x)

)

=
{4}2a`

2
cot

x

2
+

`−1∑
m=1

4({4}2am−1) D̃m(x) +
∞∑
m=`

4({4}2am−1) ϕm(x)

=
a`−1 + 2a` + a`+1

2
cot

x

2
+ E`(x) + F`(x).(2.7)

We shall make use of the representation (2.7) for x ∈ I`, and from now and till the
end of the proof of our theorem we supose that x ∈ I` but we shall not remind of
it.

The following estimate is true in view of the monotonous decay of 4({4}2am−1)
and the positivity of D̃m(x) for m ≤ `:

E`(x) ≥ 4({4}2a`−1)
`−1∑
m=1

(
1

2
cot

x

2
+ ϕm(x)

)
= 4({4}2a`−1)

(
`

2
cot

x

2
+ ψ`−1(x)

)
=
4({4}2a`−1)

4 sin2(x/2)
(` sin x− sin `x) .(2.8)



42 XHEVAT Z. KRASNIQI

Let us estimate F`(x) from above. Applying Abel’s transformation we have

|F`(x)| =

∣∣∣∣∣ limn→∞
{
n−1∑
m=`

42 ({4}2am−1) ψm(x)

+4({4}2an−1) ψn(x) −4({4}2a`−1) ψ`−1(x)

}∣∣∣∣∣
≤

∞∑
m=`

42 ({4}2am−1) |ψm(x) −ψ`−1(x)|

≤
4({4}2a`−1)

4 sin2(x/2)
(1 + sin `x) .(2.9)

From (2.8) and (2.9), in a similiar way as Telyakovskĭı did [6], for ` ≥ 11 we can
show that

1

2
E`(x) + F`(x) > 0.

Further, if m < `, then

D̃m(x) ≥
m∑
i=1

2

π
ix ≥

m(m + 1)

` + 1
>
m2

`
.

Therefore,

(2.10)
1

2
E`(x) ≥

1

2`

`−1∑
m=1

m24({4}2am−1) .

From (2.10), (2.7), and (2.6) we obtain the estimate of g(x) from below

g(x) ≥
1(

2 cos x
2

)2
(
a`−1 + 2a` + a`+1

2
cot

x

2
+

1

2`

`−1∑
m=1

m24({4}2am−1)

)
.

Since

D̃m(x) ≤ m2x ≤
πm2

`
,

then

(2.11) E`(x) ≤
π

`

`−1∑
m=1

m24({4}2am−1) .

For the estimate (2.9) we can write

|F`(x)| ≤
4({4}2a`−1)

2 sin2(x/2)
≤4({4}2a`−1)

π2

2x2
≤

(` + 1)2

2
4({4}2a`−1) ,

and for ` ≥ 11
(` + 1)2

2
<

2, 4

`

`−1∑
m=1

m2,

hence, by reason of the monotonicity of 4({4}2a`−1) we get

|F`(x)| ≤
2, 4

`

`−1∑
m=1

m24({4}2am−1) .(2.12)



ON THE BEHAVIOR NEAR THE ORIGIN OF A SINE SERIES 43

Estimates (2.12), (2.13), and (2.7) give the estimate of g(x) from above

g(x) ≤
1(

2 cos x
2

)2
(
a`−1 + 2a` + a`+1

2
cot

x

2
+

6

`

`−1∑
m=1

m24({4}2am−1)

)
.

The proof is completed. �

It follows from Theorem 2.3 that for x ∈ I` in a sufficiently small neighbourhood
of the origin we have

g(x) =
1

2 (1 + cos x)

(
{4}2a`−1

2
cot

x

2
+ O

(
1

`

`−1∑
m=1

m24({4}2am−1)

))
.(2.13)

Corollary 2.3. Assume that {am}∞m=1 is a (2, 2)-monotone sequence. Then the
following order equality is true

g(x) ∼ (`− 1){4}2a`−1 +
1

`

`−1∑
m=1

m{4}2am−1.

Proof. Since limx→0 x cot x = 1, then it is enough to prove that

1

`

`−1∑
m=1

(2m− 1){4}2am−1 − (`− 1){4}2a`−1 ≤
1

`

`−1∑
m=1

m24({4}2am−1)

and

1

`

`−1∑
m=1

m24({4}2am−1) ≤
1

`

`−1∑
m=1

(2m− 1){4}2am−1.

Indeed, putting {4}2am−1 := bm−1, we can write

1

`

`−1∑
m=1

m24bm−1 =
1

`

[
b0 + 3b1 + 5b2 + · · · + (2`− 3)b`−2 − (`− 1)2b`−1

]
≤

1

`

`−1∑
m=1

(2m− 1)bm−1 ≤
1

`

`−1∑
m=1

(2m− 1){4}2am−1,(2.14)

because by the Lemma 1.1, bm−1 ≥ 0 holds true.
On the other hand we get

(`− 1)2b`−1 ≤ `(`− 1)b`−1,
therefore the proof of the corollary is completed. �

Remark 2.1. Similar statement with Theorem 2.3 holds true for the series (1.1)
with (2, 1)-monotone coefficients.

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44 XHEVAT Z. KRASNIQI

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University of Prishtina ”Hasan Prishtina”, Faculty of Education, Department of
Mathematics and Informatics, Avenue ”Mother Theresa” 5, 10000 Prishtina, Kosova