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volume 4, issue 4, article 78, 2003.

Received 21 July, 2003;

accepted 14 November, 2003.

Communicated by:H. Bor

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Journal of Inequalities in Pure and Applied Mathematics

SOME NEW INEQUALITIES FOR TRIGONOMETRIC POLYNOMIALS WITH SPECIAL COEFFICIENTS

ŽIVORAD TOMOVSKI

Faculty of Mathematics and Natural Sciences Department of Mathematics

Skopje 1000 MACEDONIA.

EMail:tomovski@iunona.pmf.ukim.edu.mk

c

2000Victoria University ISSN (electronic): 1443-5756 101-03

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Some New Inequalities for Trigonometric Polynomials with

Special Coefficients Živorad Tomovski

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J. Ineq. Pure and Appl. Math. 4(4) Art. 78, 2003

http://jipam.vu.edu.au

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: Petrovi´c inequality, Complex trigonometric polynomial, Complex semi- convex coefficients, Complex convex coefficients.

1. Introduction and Preliminaries

Petrovi´c [4] proved the following complementary triangle inequality for se- quences of complex numbers{z₁, z₂, . . . , z_n}.

Theorem A. Let α be a real number and 0 < θ < ^π₂.If {z₁, z₂, . . . , z_n} are complex numbers such thatα−θ ≤argz_ν ≤α+θ, ν = 1,2, . . . , n,then

n

X

ν=1

z_ν

≥(cosθ)

n

X

ν=1

|z_ν|.

For0< θ < ^π₂ denote byK(θ)the coneK(θ) ={z :|argz| ≤θ}.

Let ∆λ_n = λ_n −λ_n+1, forn = 1,2,3, . . . , where {λ_n} is a sequence of complex numbers. Then,

∆²λ_n = ∆ (∆λ_n) = ∆λ_n−∆λ_n+1 =λ_n−2λ_n+1+λ_n+2, n= 1,2,3, . . . The author Tomovski (see [5]) proved the following inequality for cosine and sine polynomials with complex-valued coefficients.

Theorem B. Letx6= 2kπfork = 0,±1,±2, . . .

1. Let {bk} be a positive nondecreasing sequence and {uk} a sequence of complex numbers such that∆

uk

bk

∈K(θ).Then

m

X

k=n

u_kf(kx)

≤ 1

sin^x₂

1 + 1 cosθ

|u_m|+ 1 cosθ

b_m b_n |u_n|

, (∀n, m∈N, m > n).

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2. Let {b_k} be a positive nondecreasing sequence and {u_k} a sequence of complex numbers such that∆ (ukbk)∈K(θ).Then

m

X

k=n

u_kf(kx)

≤ 1

sin^x₂

1 + 1 cosθ

|u_n|+ 1 cosθ

b_m b_n |u_m|

, (∀n, m∈N, m > n). Heref(x) = sinxorf(x) = cosx.

Similarly, the results of TheoremBwere given by the author in [5] for sums of typePm

k=n(−1)^ku_kf(kx),where againf(x) = sinxorf(x) = cosx.

Mitrinovi´c and Peˇcari´c (see [2,3]) proved the following inequalities for cosine and sine polynomials with nonnegative coefficients.

Theorem C. Letx6= 2kπfork = 0,±1,±2, ..

1. Let {b_k} be a positive nondecreasing sequence and {a_k} a nonnegative sequence such that

a_kb⁻¹_k is a decreasing sequence. Then

m

X

k=n

a_kf(kx)

≤ a_n sin^x₂

b_m

b_n

, (∀n, m∈N, m > n).

2. Let {b_k} be a positive nondecreasing sequence and {a_k} a nonnegative sequence such that{a_kb_k}is an increasing sequence. Then

m

X

k=n

a_kf(kx)

≤ am

sin^x₂

bm

b_n

, (∀n, m∈N, m > n). Heref(x) = sinxorf(x) = cosx.

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The special cases of these inequalities were proved by G.K. Lebed forb_k = k^s, s ≥ 0(see [1]). Similarly, the results of Theorem C, were given by Mitri- novi´c and Peˇcari´c in [2,3] for sums of typePm

k=n(−1)^ka_kf(kx),where again f(x) = sinxorf(x) = cosx.

The sequence{u_k}is said to be complex semiconvex if there exists a cone K(θ), such that ∆²

uk

bk

∈ K(θ) or ∆²(ukbk) ∈ K(θ), where {bk} is a positive nondecreasing sequence. Forb_k= 1, the sequence{u_k}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.

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2. Main Results

Theorem 2.1. Let {z_k} be a sequence of complex numbers such that A =

n≤p≤q≤mmax

Pq j=p

Pj k=izk

. Further, let {bk} be a positive nondecreasing se- quence. If{u_k}is a sequence of complex numbers such that∆²

uk

bk

∈K(θ), then

m

X

k=n

u_kz_k

≤A

|u_m|+b_m

1 + 1 cosθ

∆

um−1

bm−1

+ b_m cosθ

∆ u_n

b_n

, (∀n, m∈N, m > n). Proof. Let us estimate the sumPm

k=nb_kz_k. Since

m

X

k=n

z_k

≤

m

X

j=n+1

j

X

k=n

z_k

≤A, we obtain

m

X

k=n

b_kz_k

=

b_n

m

X

k=n

z_k+

m

X

j=n+1 m

X

k=j

z_k

!

(b_j −bj−1)

≤bn

m

X

k=n

zk

+

m

X

j=n+1

m

X

k=j

zk

(bj−bj−1)

≤A(b_n+b_m−b_n) =Ab_m. (*)

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Then,

m

X

k=n

u_kz_k

=

m

X

k=n

u_k bk

(b_kz_k)

=

u_m b_m

m

X

k=n

b_kz_k+

m−1

X

j=n j

X

k=n

b_kz_k

!

∆ u_j

b_j

=

um

b_m

m

X

k=n

b_kz_k+ ∆

um−1

bm−1

^m−1 X

j=n j

X

k=n

b_kz_k

+

m−2

X

r=n

∆² u_r

b_r ^r

X

j=n j

X

k=n

b_kz_k

≤ |u_m| b_m

m

X

k=n

b_kz_k

+

∆

um−1

b_m−1

m−1

X

j=n j

X

k=n

b_kz_k

+

m−2

X

r=n

∆² u_r

b_r

r

X

j=n j

X

k=n

b_kz_k

≤Abm

|u_m|

b_m +Abm

∆

u_m−1 bm−1

+ Ab_m cosθ

m−2

X

r=n

∆² u_r

b_r

=A

|u_m|+b_m

∆

um−1

bm−1

+ b_m cosθ

∆ u_n

b_n

−∆

um−1

bm−1

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≤A

|u_m|+b_m

1 + 1 cosθ

∆

um−1

bm−1

+ b_m cosθ

∆ u_n

b_n

.

Theorem 2.2. Let {z_k} and {b_k} be defined as in Theorem 2.1. If {u_k} is a sequence of complex numbers such that∆²(u_kb_k)∈K(θ),then

m

X

k=n

u_kz_k

≤A

|u_n|+b⁻¹_n

1 + 1 cosθ

(|∆ (u_nb_n)|+|∆ (um−1bm−1)|)

, (∀n, m∈N, m > n). Proof. The sequence

b⁻¹_k ^m

k=nis nonincreasing, so from (*) we get

m

X

k=n

b⁻¹_k z_k

≤Ab⁻¹_n . Now, we have:

m

X

k=n

u_kz_k

=

m

X

k=n

(u_kb_k)b⁻¹_k z_k

=

u_nb_n

m

X

k=n

b⁻¹_k z_k+

m

X

j=n+1 m

X

k=j

b⁻¹_k z_k

!

(u_jb_j −uj−1bj−1)

=

u_nb_n

m

X

k=n

b⁻¹_k z_k−

m−1

X

j=n+1

∆²(uj−1bj−1)

j

X

r=n m

X

k=r

b⁻¹_k z_k

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+ ∆ (u_nb_n)

m

X

k=n

b⁻¹_k z_k−∆ (um−1bm−1)

m

X

r=n m

X

k=r

b⁻¹_k z_k

≤ |u_n|b_n

m

X

k=n

b⁻¹_k z_k

+

m−1

X

j=n+1

∆²(uj−1bj−1)

j

X

r=n m

X

k=r

b⁻¹_k z_k

+|∆ (u_nb_n)|

m

X

k=n

b⁻¹_k z_k

+|∆ (um−1bm−1)|

m

X

r=n m

X

k=r

b⁻¹_k z_k

≤ |u_n|b_nAb⁻¹_n +Ab⁻¹_n

m−1

X

j=n+1

∆²(uj−1bj−1)

+Ab⁻¹_n |∆ (u_nb_n)|

+Ab⁻¹_n |∆ (um−1bm−1)|

≤A

"

|un|+ b⁻¹_n cosθ

m−1

X

j=n+1

∆²(uj−1bj−1)

+b⁻¹_n |∆ (u_nb_n)|+b⁻¹_n |∆ (u_m−1b_m−1)|

=A

|u_n|+ b⁻¹_n

cosθ |∆ (u_nb_n)−∆ (um−1bm−1)|

+ b⁻¹_n |∆ (u_nb_n)|+b⁻¹_n |∆ (um−1bm−1)|

≤A

|un|+b⁻¹_n

1 + 1 cosθ

(|∆ (unbn)|+|∆ (um−1bm−1)|)

.

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Lemma 2.3. For allp, q ∈N,p < q, the following inequalities hold

(2.1)

q

X

j=p j

X

k=l

e^ikx

≤ q−p+ 2

2 sin² ^x₂ , x6= 2kπ, k = 0,±1,±2, . . . ,

(2.2)

q

X

j=p j

X

k=l

(−1)^ke^ikx

≤ q−p+ 2

2 cos² ^x₂ , x6= (2k+ 1)π,

k = 0,±1,±2, . . . . Proof. It is sufficient to prove the first inequality, since the second inequality can be proved analogously.

q

X

j=p j

X

k=l

e^ikx

=

q

X

j=p

eîlxeî(j−l+1)x−1 eîx−1

= 1

|e^ix−1|

1 e^i(l−1)x

q

X

j=p

e^ijx−(q−p+ 1)

≤ 1

2 sin^x₂

e^i(q−p+1)−1

|e^ix −1| +q−p+ 1 2 sin^x₂

≤ 2

4 sin² ^x₂ +q−p+ 1

2 sin² ^x₂ = q−p+ 2 2 sin² ^x₂ .

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By puttingz_k = exp (ikx)in Theorem 2.1and Theorem 2.2 and using the inequality (2.1) of the above lemma, we have:

Theorem 2.4. (i) Let{b_k}and{u_k}be defined as in Theorem2.1. Then

m

X

k=n

u_kexp (ikx)

≤ m−n+ 2 2 sin² ^x₂

|u_m|+b_m

1 + 1 cosθ

∆

um−1

bm−1

+ b_m cosθ

∆ u_n

b_n

, (∀n, m∈N, m > n). (ii) Let{b_k}and{u_k}be defined as in Theorem2.2. Then

m

X

k=n

u_kexp (ikx)

≤ m−n+ 2 2 sin² ^x₂

|u_n|+b⁻¹_n

1 + 1 cosθ

(|∆ (u_nb_n)|+|∆ (u_m−1b_m−1)|)

, (∀n, m∈N, m > n). In both casesx6= 2kπ, k = 0,±1,±2, . . .

Applying the known inequalitiesRez ≤ |z|and Imz ≤ |z| forz ∈ C, we obtain the following result:

Theorem 2.5. Letx6= 2kπfork= 0,±1,±2, . . . .

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(i) Let{b_k}and{u_k}be defined as in Theorem2.1. Then

m

X

k=n

u_kf(kx)

≤ m−n+ 2 2 sin² ^x₂

|u_m|+b_m

1 + 1 cosθ

∆

um−1

b_m−1

+ b_m cosθ

∆ u_n

b_n

, (∀n, m∈N, m > n). (ii) Let{b_k}and{u_k}be defined as in Theorem2.2. Then

m

X

k=n

u_kf(kx)

≤ m−n+ 2 2 sin² ^x₂

|u_n|+b⁻¹_n

1 + 1 cosθ

(|∆ (u_nb_n)|+|∆ (u_m−1b_m−1)|)

, (∀n, m∈N, m > n). Applying inequality (2.2) of Lemma2.3, we obtain the following results:

Theorem 2.6. Letx6= (2k+ 1)πfork = 0,±1,±2, . . . and letx7→f(x) be defined as in Theorem2.5.

(i) If{b_k}and{u_k}are defined as in Theorem2.1, then

m

X

k=n

(−1)^ku_kf(kx)

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≤ m−n+ 2 2 cos² ^x₂

|u_m|+b_m

1 + 1 cosθ

∆

um−1

bm−1

+ b_m cosθ

∆ u_n

b_n

, (∀n, m∈N, m > n). (ii) If{b_k}and{u_k}are defined as in Theorem2.2, then

m

X

k=n

(−1)^ku_kf(kx)

≤ m−n+ 2 2 cos² ^x₂

|u_n|+b⁻¹_n

1 + 1 cosθ

(|∆ (u_nb_n)|+|∆ (um−1bm−1)|)

, (∀n, m∈N, m > n). Forb_k = 1,we obtain the following theorem.

Theorem 2.7. Let{u_k}be a complex convex sequence.

(i) Ifx6= 2kπfork = 0,±1,±2, . . ., then we have:

m

X

k=n

u_kf(kx)

≤ m−n+ 2 2 sin² ^x₂

|u_m|+

1 + 1 cosθ

|∆um−1|+ 1

cosθ|∆u_n|

, (∀n, m∈N, m > n).

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(ii) Ifx6= (2k+ 1)πfork = 0,±1,±2, . . ., then we have:

m

X

k=n

(−1)^ku_kf(kx)

≤ m−n+ 2 2 cos² ^x₂

|un|+

1 + 1 cosθ

(|∆un|+|∆um−1|)

, (∀n, m∈N, m > n).

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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. MITRINOVI ´CANDJ.E. PE ˇCARI ´C, On an inequality of G. K. Lebed’, Prilozi MANU, Od. Mat. Tehn. Nauki, 12 (1991), 15–19

[3] J.E. PE ˇCARI ´C, Nejednakosti, 6 (1996), 175–179 (Zagreb).

[4] M. PETROVI ´C, Theoreme sur les integrales curvilignes, Publ. Math. Univ.

Belgrade, 2 (1933), 45–59

[5] Ž. TOMOVSKI, On some inequalities of Mitrinovi´c and Peˇcari´c, Prilozi MANU, Od. Mat. Tehn. Nauki, 22 (2001), 21–28.