Journal of Inequalities in Pure and
Applied Mathematics
SOME NEW INEQUALITIES FOR TRIGONOMETRIC
POLYNOMIALS WITH SPECIAL COEFFICIENTS
volume 4, issue 4, article 78,
2003.
ŽIVORAD TOMOVSKI
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Skopje 1000 MACEDONIA.
EMail: tomovski@iunona.pmf.ukim.edu.mk
Received 21 July, 2003;
accepted 14 November, 2003.
Communicated by: H. Bor
Abstract
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2000
Victoria University
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Abstract
Some new inequalities for certain trigonometric polynomials with complex semiconvex and complex convex coefficients are given.
2000 Mathematics Subject Classification: 26D05, 42A05
Key words: Petrović inequality, Complex trigonometric polynomial, Complex semiconvex coefficients, Complex convex coefficients.
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Contents
1
Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
Živorad Tomovski
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1.
Introduction and Preliminaries
Petrović [4] proved the following complementary triangle inequality for sequences of complex numbers {z1 , z2 , . . . , zn } .
Theorem A. Let α be a real number and 0 < θ < π2 . If {z1 , z2 , . . . , zn } are
complex numbers such that α − θ ≤ arg zν ≤ α + θ, ν = 1, 2, . . . , n, then
n
n
X
X
z
≥
(cos
θ)
|zν |.
ν
ν=1 ν=1
For 0 < θ < π2 denote by K (θ) the cone K (θ) = {z : |arg z| ≤ θ} .
Let ∆λn = λn − λn+1 , for n = 1, 2, 3, . . . , where {λn } is a sequence of
complex numbers. Then,
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
∆2 λn = ∆ (∆λn ) = ∆λn − ∆λn+1 = λn − 2λn+1 + λn+2 , n = 1, 2, 3, . . .
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The author Tomovski (see [5]) proved the following inequality for cosine and
sine polynomials with complex-valued coefficients.
Contents
Theorem B. Let x 6= 2kπ for k = 0, ±1, ±2, . . .
1. Let {bk } be a positive nondecreasing
sequence and {uk } a sequence of
complex numbers such that ∆ ubkk ∈ K (θ) . Then
m
X
1
1 bm
1 |um | +
|un | ,
uk f (kx) ≤ x 1+
cos θ
cos θ bn
sin 2
k=n
(∀n, m ∈N, m > n) .
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2. Let {bk } be a positive nondecreasing sequence and {uk } a sequence of
complex numbers such that ∆ (uk bk ) ∈ K (θ) . Then
m
X
1
1 bm
1 uk f (kx) ≤ x 1+
|un | +
|um | ,
cos
θ
cos
θ
b
sin
n
2
k=n
(∀n, m ∈N, m > n) .
Here f (x) = sin x or f (x) = cos x.
Similarly,
of Theorem B were given by the author in [5] for sums
Pm the results
k
of type k=n (−1) uk f (kx) , where again f (x) = sin x or f (x) = cos x.
Mitrinović and Pečarić (see [2, 3]) proved the following inequalities for cosine and sine polynomials with nonnegative coefficients.
Theorem C. Let x 6= 2kπ for k = 0, ±1, ±2, ..
1. Let {bk } be a positive
sequence and {ak } a nonnegative
nondecreasing
−1
sequence such that ak bk is a decreasing sequence. Then
m
X
an b m
ak f (kx) ≤ x , (∀n, m ∈N, m > n) .
bn
sin 2
k=n
2. Let {bk } be a positive nondecreasing sequence and {ak } a nonnegative
sequence such that {ak bk } is an increasing sequence. Then
m
X
am b m
ak f (kx) ≤ x , (∀n, m ∈N, m > n) .
b
sin
n
2
k=n
Heref (x) = sin x or f (x) = cos x.
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
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The special cases of these inequalities were proved by G.K. Lebed for bk =
k , s ≥ 0 (see [1]). Similarly, the results of
PTheorem kC, were given by Mitrinović and Pečarić in [2, 3] for sums of type m
k=n (−1) ak f (kx) , where again
f (x) = sin x or f (x) = cos x.
The sequence {uk
} is said to be complex semiconvex if there exists a cone
2 uk
K (θ), such that ∆ bk ∈ K (θ) or ∆2 (uk bk ) ∈ K (θ) , where {bk } is a
positive nondecreasing sequence. For bk = 1, the sequence {uk } shall be called
a complex convex sequence.
In this paper we shall give some estimates for cosine and sine polynomials
with complex semi-convex and complex convex coefficients.
s
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
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2.
Main Results
Theorem 2.1. Let {zk } be a sequence of complex numbers such that A =
P P
max qj=p jk=i zk . Further, let {bk } be a positive nondecreasing sen≤p≤q≤m
quence. If {uk } is a sequence of complex numbers such that ∆2 ubkk ∈ K (θ) ,
then
m
X
uk zk k=n
um−1 bm un 1
≤ A |um | + bm 1 +
∆
+
∆
,
cos θ
bm−1
cos θ
bn (∀n, m ∈N, m > n) .
Pm
Proof. Let us estimate the sum k=n bk zk .
Since
m j
m X
X X
zk ≤
zk ≤ A,
j=n+1 k=n
k=n
we obtain
m
m
m
X
X
X
bk zk = bn
zk +
j=n+1
k=n
k=n
m
m
X
X
≤ bn zk +
k=n
(*)
m
X
!
zk
k=j
(bj − bj−1 )
m
X
zk (bj − bj−1 )
Some New Inequalities for
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Živorad Tomovski
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j=n+1 k=j
≤ A (bn + bm − bn ) = Abm .
J. Ineq. Pure and Appl. Math. 4(4) Art. 78, 2003
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Then,
m
m
X u
X
k
uk zk = (bk zk )
bk
k=n
k=n
! j
m
m−1
u X
X
X
uj m
=
bk zk +
bk zk ∆
bm
b
j
j=n
k=n
k=n
m−1 j
m
u X
um−1 X X
m
=
bk zk + ∆
bk zk
bm
bm−1 j=n k=n
k=n
X
j
r X
m−2
X
u
r
+
∆2
bk zk br j=n k=n
r=n
m−1 j
m
um−1 X X
|um | X
≤
bk zk + ∆
b
z
k k
bm k=n
bm−1 j=n k=n
j
m−2
r X
X ur X
2
bk zk +
∆ br r=n
j=n k=n
m−2
X
u
Ab
ur |um |
m−1 m + Abm ∆
+
≤ Abm
∆2
bm
bm−1
cos θ r=n
br um−1 bm un
um−1 = A |um | + bm ∆
+
∆
−∆
bm−1 cos θ bn
bm−1 Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
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≤ A |um | + bm 1 +
1
cos θ
∆ um−1 + bm ∆ un .
bm−1 cos θ bn Theorem 2.2. Let {zk } and {bk } be defined as in Theorem 2.1. If {uk } is a
sequence of complex numbers such that ∆2 (uk bk ) ∈ K (θ) , then
m
X
1
−1
uk zk ≤ A |un | + bn 1 +
(|∆ (un bn )| + |∆ (um−1 bm−1 )|) ,
cos θ
k=n
(∀n, m ∈N, m > n) .
m
Proof. The sequence b−1
is nonincreasing, so from (*) we get
k
k=n
m
X
−1 bk zk ≤ Ab−1
n .
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
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k=n
Now, we have:
m
m
X
X
−1 uk zk = (uk bk ) bk zk k=n
k=n
!
m
m
m
X
X
X
−1
−1
bk zk (uj bj − uj−1 bj−1 )
= un bn
bk zk +
j=n+1
k=n
k=j
j
m
m−1
m
X
X
X
X
−1
2
= un bn
bk zk −
∆ (uj−1 bj−1 )
b−1
k zk
r=n
k=n
j=n+1
k=r
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+ ∆ (un bn )
m
X
b−1
k zk − ∆ (um−1 bm−1 )
m X
m
X
r=n k=r
k=n
−1 bk zk j
m
m−1
m
X
X X
X −1 2
∆
(u
b
)
≤ |un | bn b−1
z
+
b
z
j−1 j−1
k
k
k
k
r=n k=r
j=n+1
k=n
m
m X
m
X
X
−1 + |∆ (un bn )| b−1
z
+
|∆
(u
b
)|
z
b
k
m−1 m−1 k
k
k
r=n k=r
k=n
≤
|un | bn Ab−1
n
+
Ab−1
n
m−1
X
2
∆ (uj−1 bj−1 ) + Ab−1
n |∆ (un bn )|
j=n+1
Živorad Tomovski
Ab−1
n
|∆ (um−1 bm−1 )|
−1 m−1
X
bn 2
≤ A |un | +
∆ (uj−1 bj−1 )
cos θ +
"
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
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j=n+1
−1
+ b−1
n |∆ (un bn )| + bn |∆ (um−1 bm−1 )|
b−1
= A |un | + n |∆ (un bn ) − ∆ (um−1 bm−1 )|
cos θ
−1
+ b−1
n |∆ (un bn )| + bn |∆ (um−1 bm−1 )|
1
−1
≤ A |un | + bn 1 +
(|∆ (un bn )| + |∆ (um−1 bm−1 )|) .
cos θ
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Lemma 2.3. For all p, q ∈ N, p < q, the following inequalities hold
q j
X X
q−p+2
ikx (2.1)
, x 6= 2kπ, k = 0, ±1, ±2, . . . ,
e
≤
2 sin2 x2
j=p k=l
(2.2)
q j
X X
q−p+2
, x 6= (2k + 1) π,
(−1)k eikx ≤
2 cos2 x2
j=p k=l
k = 0, ±1, ±2, . . . .
Proof. It is sufficient to prove the first inequality, since the second inequality
can be proved analogously.
q j
q
X X
X
i(j−l+1)x
e
−
1
eikx = eilx
ix − 1
e
j=p k=l
j=p
q
X
1
1
ijx
= ix
e − (q − p + 1)
i(l−1)x
|e − 1| e
j=p
i(q−p+1)
e
− 1 q − p + 1
1
≤ + 2 sin x2 |eix − 1|
2 sin x2 2
q−p+1
q−p+2
≤
.
2 x +
2 x =
4 sin 2
2 sin 2
2 sin2 x2
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Živorad Tomovski
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By putting zk = exp (ikx) in Theorem 2.1 and Theorem 2.2 and using the
inequality (2.1) of the above lemma, we have:
Theorem 2.4.
(i) Let {bk } and {uk } be defined as in Theorem 2.1. Then
m
X
uk exp (ikx)
k=n
u
b
m−n+2
1
m−1
m
∆
+
∆ un ,
≤
|um | + bm 1 +
2 x
cos θ
bm−1
cos θ
bn 2 sin 2
(∀n, m ∈N, m > n) .
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
(ii) Let {bk } and {uk } be defined as in Theorem 2.2. Then
m
X
uk exp (ikx)
k=n
m−n+2
1
−1
≤
|un | + bn 1 +
(|∆ (un bn )| + |∆ (um−1 bm−1 )|) ,
cos θ
2 sin2 x2
(∀n, m ∈N, m > n) .
In both cases x 6= 2kπ, k = 0, ±1, ±2, . . .
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Applying the known inequalities Re z ≤ |z| and Im z ≤ |z| for z ∈ C, we
obtain the following result:
Theorem 2.5. Let x 6= 2kπ for k = 0, ±1, ±2, . . . .
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(i) Let {bk } and {uk } be defined as in Theorem 2.1. Then
m
X
u
f
(kx)
k
k=n
m−n+2
1
u
b
m−1
m
+
∆
∆ un ,
≤
|u
|
+
b
1
+
m
m
cos θ bm−1 cos θ bn 2 sin2 x2
(∀n, m ∈N, m > n) .
(ii) Let {bk } and {uk } be defined as in Theorem 2.2. Then
m
X
uk f (kx)
k=n
m−n+2
1
−1
≤
(|∆ (un bn )| + |∆ (um−1 bm−1 )|) ,
|un | + bn 1 +
cos θ
2 sin2 x2
(∀n, m ∈N, m > n) .
Applying inequality (2.2) of Lemma 2.3, we obtain the following results:
Theorem 2.6. Let x 6= (2k + 1) π for k = 0, ±1, ±2, . . . and let x 7→ f (x) be
defined as in Theorem 2.5.
(i) If {bk } and {uk } are defined as in Theorem 2.1, then
m
X
k
(−1) uk f (kx)
k=n
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
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m−n+2
1
um−1 bm un +
≤
|u
|
+
b
1
+
∆
∆
,
m
m
2 cos2 x2
cos θ bm−1 cos θ bn (∀n, m ∈N, m > n) .
(ii) If {bk } and {uk } are defined as in Theorem 2.2, then
m
X
(−1)k uk f (kx)
k=n
1
m−n+2
−1
|un | + bn 1 +
(|∆ (un bn )| + |∆ (um−1 bm−1 )|) ,
≤
cos θ
2 cos2 x2
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
(∀n, m ∈N, m > n) .
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For bk = 1, we obtain the following theorem.
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Theorem 2.7. Let {uk } be a complex convex sequence.
(i) If x 6= 2kπ for k = 0, ±1, ±2, . . . , then we have:
m
X
uk f (kx)
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k=n
m−n+2
1
1
|∆um−1 | +
|∆un | ,
≤
|um | + 1 +
cos θ
cos θ
2 sin2 x2
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(∀n, m ∈N, m > n) .
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(ii) If x 6= (2k + 1) π for k = 0, ±1, ±2, . . . , then we have:
m
X
(−1)k uk f (kx)
k=n
m−n+2
1
≤
|un | + 1 +
(|∆un | + |∆um−1 |) ,
2 cos2 x2
cos θ
(∀n, m ∈N, m > n) .
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
Živorad Tomovski
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References
[1] G.K. LEBED, On trigonometric series with coefficients which satisfy some
conditions (Russian), Mat. Sb., 74 (116) (1967), 100–118
[2] D.S. MITRINOVIĆ AND J.E. PEČARIĆ, On an inequality of G. K. Lebed’,
Prilozi MANU, Od. Mat. Tehn. Nauki, 12 (1991), 15–19
[3] J.E. PEČARIĆ, Nejednakosti, 6 (1996), 175–179 (Zagreb).
[4] M. PETROVIĆ, Theoreme sur les integrales curvilignes, Publ. Math. Univ.
Belgrade, 2 (1933), 45–59
Some New Inequalities for
Trigonometric Polynomials with
Special Coefficients
[5] Ž. TOMOVSKI, On some inequalities of Mitrinović and Pečarić, Prilozi
MANU, Od. Mat. Tehn. Nauki, 22 (2001), 21–28.
Živorad Tomovski
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