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volume 6, issue 4, article 98, 2005.

Received 08 April, 2005;

accepted 03 September, 2005.

Communicated by:D. Stefanescu

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Journal of Inequalities in Pure and Applied Mathematics

INEQUALITIES FOR WALSH POLYNOMIALS WITH SEMI-MONOTONE AND SEMI-CONVEX COEFFICIENTS

ŽIVORAD TOMOVSKI

University "St. Cyril and Methodius"

Faculty of Natural Sciences and Mathematics Institute of Mathematics

PO Box 162, 91000 Skopje Repubic of Macedonia.

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

c

2000Victoria University ISSN (electronic): 1443-5756 112-05

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Inequalities for Walsh Polynomials with Semi-Monotone and Semi-Convex Coefficients

Živorad Tomovski

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J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005

http://jipam.vu.edu.au

Abstract

Using the concept of majorant sequences (see [4, ch. XXI], [5], [7], [8]) some new inequalities for Walsh polynomials with complex semi-monotone, complex semi-convex, complex monotone and complex convex coefficients are given.

2000 Mathematics Subject Classification:26D05, 42C10.

Key words: Petrovic inequality, Walsh polynomial, Complex semi-convex coeffi- cients, Complex convex coefficients, Complex semi-monotone coefficients, Complex monotone coefficients, Fine inequality.

1. Introduction and Preliminaries

We consider the Walsh orthonormal system{w_n(x)}^∞_n=0 defined on[0,1)in the Paley enumeration. Thus w₀(x) ≡ 1and for each positive integer with dyadic development

n=

p

X

i=1

2^νⁱ, ν1 > ν2 >· · ·> νp ≥0, we have

w_n(x) =

p

Y

i=1

r_ν_i(x),

where{r_n(x)}^∞_n=0denotes the Rademacher system of functions defined by (see, e.g. [1, p. 60], [3, p. 9-10])

rν(x) = sign sin 2^νπ(x) (ν = 0,1,2, . . .; 0≤x <1).

In this paper we shall consider the Walsh polynomialsPm

k=nλ_kw_k(x) with complex-valued coefficients{λ_k}.

Let ∆λ_n = λ_n −λ_n+1 and ∆²λ_n = ∆(∆λ_n) = ∆λ_n−∆λ_n+1 = λ_n− 2λn+1+λn+2, for alln = 1,2,3. . ..

Petrovi´c [6] proved the following complementary triangle inequality for a sequence of complex numbers{z₁, z₂, . . . , z_n}.

Theorem A. Let α be a real number and0 < θ < ^π₂. If{z1, z2, . . . , zn} 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|.

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

Let {b_k} be a positive nondecreasing sequence. The following definitions are given in [7] and [8]. The sequence of complex numbers {u_k} is said to be complex semi−monotone if there exists a coneK(θ) such that∆

uk

bk

∈ K(θ) or ∆(u_kb_k) ∈ K(θ). For b_k = 1, the sequence {u_k} shall be called a complex monotone sequence. On the other hand, the sequence {uk} is said to be complex semi−convex if there exists a coneK(θ), such that∆²

uk

b_k

∈ K(θ) or ∆²(ukbk) ∈ K(θ). For bk = 1, the sequence {uk} shall be called a complex convex sequence.

The following two Theorems were proved by Tomovski in [7] and [8].

Theorem B ([7]). Let{z_k}be a sequence such that|Pm

k=nz_k| ≤ A, (∀n, m∈ N, m > n),whereAis a positive number.

(i) If∆

uk

b_k

∈K(θ),then

m

X

k=n

u_kz_k

≤A

1 + 1 cosθ

|u_m|+ 1 cosθ

b_m bn

|u_n|

, (∀n, m∈N, m > n).

(ii) If∆(u_kb_k)∈K(θ), then

m

X

k=n

u_kz_k

≤A

1 + 1 cosθ

|u_n|+ 1 cosθ

bm

b_n|u_m|

, (∀n, m∈N, m > n).

Theorem C ([8]). LetA= max

n≤p≤q≤m

q

P

j=p j

P

k=i

zk

.

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(i) If{uk}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

b_m−1

+ b_m cosθ

∆ u_n

b_n

, (∀n, m∈N, m > n).

(ii) 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)|+|∆(u_m−1b_m−1)|)

, (∀n, m∈N, m > n).

Using the concept of majorant sequences we shall give some estimates for Walsh polynomials with complex semi-monotone, complex monotone, complex semi-convex and complex convex coefficients.

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2. Main Results

For the main results we require the following Lemma.

Lemma 2.1. For allp, q, r∈N, p < qthe following inequalities hold:

q

X

k=p

ω_k(x)

≤ 2

x,0< x <1.

(i)

q

X

j=p j

X

k=l

ω_k(x) (ii)

≤







2(q−p+1)

x =C1(p, q, x) : 0 < x <1

8

x(x−2^−r)+ _x⁸2 + ^2(q−p+1)_x + 1 =C₂(p, q, r, x) :x∈(2^−r,2^−r+1) Proof. (i) Let D_q(x) = Pq−1

i=0 w_i(x) be the Dirichlet kernel. Then it is known that (see [3, p. 28])|D_q(x)| ≤ _x¹,0< x <1. Hence

q

X

k=p

w_k(x)

=|D_q+1(x)−D_p(x)| ≤ |D_q+1(x)|+|D_p(x)| ≤ 2 x. (ii) By (i) we get

q

X

j=p j

X

k=l

wk(x)

≤

q

X

j=p

j

X

k=l

wk(x)

≤ 2(q−p+ 1)

x , 0< x <1.

LetF_n(x) = _n+1¹ Pn

k=0D_k(x)be the Fejer kernel. Applying Fine’s inequality (see [2])

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(n+ 1)Fn(x)< 4

x(x−2^−r)+ 4

x², x∈(2^−r,2^−r+1), we get

q

X

j=p j

X

k=l

w_k(x)

=

q

X

j=p

(D_j+1(x)−D_l(x))

≤

q

X

j=p

D_j+1(x)

+q−p+ 1 x

≤ |(q+ 1)F_q(x)|+|D_q+1(x)|+|D₀(x)|

+|pF_p−1(x)|+ q−p+ 1 x

< 8

x(x−2^−r)+ 8

x² + 2(q−p+ 1)

x + 1, x∈(2^−r,2^−r+1).

Applying the inequality (i) of the above lemma and TheoremB, we obtain following theorem.

Theorem 2.2. Let0< x <1.

(i) If{u_k}is a sequence of complex numbers such that∆

uk

b_k

∈K(θ), then

m

X

k=n

u_kw_k(x)

≤ 2 x

1 + 1 cosθ

|u_m|+ 1 cosθ

b_m b_n|u_n|

, (∀n, m∈N, m > n).

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(ii) If{u_k}is a sequence of complex numbers such that∆(u_kb_k)∈K(θ), then

m

X

k=n

u_kw_k(x)

≤ 2 x

1 + 1 cosθ

|u_n|+ 1 cosθ

b_m b_n|u_m|

, (∀n, m∈N, m > n).

Specially forb_k= 1we get the following inequalities for Walsh polynomials with complex monotone coefficients.

Corollary 2.3. Let0< x <1. If{u_k}is a sequence of complex numbers such that∆u_k ∈K(θ), then

m

X

k=n

u_kω_k(x)

≤ 2 x

1 + 1 cosθ

|u_m|+ 1 cosθ|u_n|

, (∀n, m∈N, m > n).

Corollary 2.4. Let 0 < x < 1. If{u_k}is a complex monotone sequence such that lim

k→∞uk = 0, then

∞

X

k=n

u_kω_k(x)

≤ 2

xcosθ|u_n|.

In [4] (chapter XXI), [5] Mitrinovi´c and Peˇcari´c obtained inequalities for cosine and sine polynomials with monotone nonnegative coefficients. Applying Theorem 2.2, we get analogical results for Walsh polynomials with monotone nonnegative coefficients.

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Corollary 2.5. Let0< x <1.

(i) If {a_k} is a nonnegative sequence such that {a_kb⁻¹_k } is a decreasing se- quence, then

m

X

k=n

a_kw_k(x)

≤ a_n x

b_m b_n

, (∀n, m∈N, m > n).

(ii) If {a_k} is a nonnegative sequence such that {a_kb_k} is an increasing se- quence, then

m

X

k=n

a_kw_k(x)

≤ a_m x

b_m b_n

, (∀n, m∈N, m > n).

Now, applying the inequality (ii) of Lemma2.1, we obtain new inequalities for Walsh polynomials with complex semi-convex coefficients.

Theorem 2.6.

(i) If{u_k}is a sequence of complex numbers such that∆²

uk

bk

∈K(θ), then

m

X

k=n

u_kw_k(x)

≤











C₁(m, n, x)h

|u_m|+bm−1 1 + _cos¹_θ ∆_u

m−1

bm−1

+^b_cos^m−2_θ

∆

un

bn

i

: 0< x < 1 C₂(m, n, r, x)h

|u_m|+b_m−1 1 + _cos¹_θ ∆

um−1

bm−1

+^b_cos^m−2_θ

∆

un

bn

i

:x∈(2^−r,2^−r+1) for all n, m, r ∈N, m > n.

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(ii) If {u_k} is a sequence of complex numbers such that ∆²(u_kb_k) ∈ K(θ), then

m

X

k=n

ukwk(x)

≤











C₁(m, n, x)

|u_n|+b⁻¹_n 1 + _cos¹_θ

×(|∆(u_nb_n)|+|∆(u_m−1b_m−1)|)] : 0< x <1 C₂(m, n, r, x)

|u_n|+b⁻¹_n 1 + _cos¹_θ

×(|∆(u_nb_n)|+|∆(um−1bm−1)|)] :x∈(2^−r,2^−r+1) for all n, m, r ∈N, m > n.

Proof. (i) Applying Abel’s transformation twice and the triangle inequality, we get:

m

X

k=n

u_k bk

(b_kw_k)

=

u_m bm

m

X

k=n

b_kw_k+ ∆

um−1

bm−1

^m−1 X

j=n j

X

k=n

b_kw_k

+

m−2

X

r=n

∆² u_r

b_r r

X

j=n j

X

k=n

b_kw_k

≤ |um| b_m b_m

m

X

k=n

w_k

+bm−1

∆

um−1

bm−1

m−1

X

j=n j

X

k=n

w_k

+b_m−2

m−2

X

r=n

∆² u_r

b_r

r

X

j=n j

X

k=n

w_k .

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Using the Petrovi´c inequality and inequality (ii) of Lemma2.1, we obtain:

m

X

k=n

u_kw_k(x)

≤ |u_m|

m

X

k=n

w_k

+bm−1

∆

um−1

b_m−1

m−1

X

j=n j

X

k=n

w_k

+bm−2

cosθ

m−2

X

r=n

∆² ur

b_r ^r

X

j=n j

X

k=n

w_k

≤











C₁(m, n, x)h

|u_m|+bm−1 1 + _cos¹_θ ∆

um−1

bm−1

+^b_cos^m−2_θ

∆

un

bn

i

: 0< x <1 C₂(m, n, r, x)h

|u_m|+bm−1 1 + _cos¹_θ ∆

um−1

bm−1

+^b_cos^m−2_θ

∆

un

bn

i

:x∈(2^−r,2^−r+1) (ii) Analogously as the proof of (i), we obtain:

m

X

k=n

(u_kb_k)b⁻¹_k w_k

=

u_nb_n

m

X

k=n

b⁻¹_k w_k−

m−1

X

j=n+1

∆²(uj−1bj−1)

j

X

r=n m

X

k=r

b⁻¹_k w_k

+∆(u_nb_n)

m

X

k=n

b⁻¹_k w_k−∆(um−1bm−1)

m

X

r=n m

X

k=r

b⁻¹_k w_k

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≤ |u_n|b_nb⁻¹_n

m

X

k=n

w_k

+b⁻¹_n

m−1

X

j=n+1

∆²(uj−1bj−1)

j

X

r=n m

X

k=r

w_k +b⁻¹_n |∆(u_nb_n)|

m

X

k=n

w_k

+b⁻¹_n |∆(um−1bm−1)|

m

X

r=n m

X

k=r

w_k .

Hence,

m

X

k=n

u_kw_k(x)

≤ |u_n|

m

X

k=n

w_k

+ b⁻¹_n cosθ

m−1

X

j=n+1

∆²(uj−1bj−1)

j

X

r=n m

X

k=r

w_k +b⁻¹_n |∆(u_nb_n)|

m

X

k=n

w_k

+b⁻¹_n |∆(um−1bm−1)|

m

X

r=n m

X

k=r

w_k

≤











C₁(m, n, x)

|u_n|+b⁻¹_n 1 + _cos¹_θ

×(|∆(u_nb_n)|+|∆(um−1bm−1)|)] : 0 < x <1, C₂(m, n, r, x)

|u_n|+b⁻¹_n 1 + _cos¹_θ

×(|∆(u_nb_n)|+|∆(um−1bm−1)|)] :x∈(2^−r,2^−r+1).

Ifbk = 1, k = n, n+ 1, . . . , mfrom Theorem2.6, we obtain the following corollary.

Corollary 2.7. Let{u_k}be a complex-convex sequence. Then,

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m

X

k=n

u_kw_k(x)

≤











C₁(m, n, x)

|u_m|+ 1 + _cos¹_θ

×|∆um−1|+_cos¹_θ|∆un|

: 0< x <1 C2(m, n, r, x)

|um|+ 1 + _cos¹_θ

×|∆um−1|+_cos¹_θ|∆u_n|

Remark 1. Similarly, the results of Theorem 2.2, Theorem 2.6, Corollary 2.3, Corollary2.5and Corollary2.7were given by the author in [7,8] for trigono- metric polynomials with complex valued coefficients.

Corollary 2.8.

(i) If{a_k}is a nonnegative sequence such that{a_kb⁻¹_k }is a convex sequence, then

m

X

k=n

a_kw_k(x)

≤











C₁(m, n, x)h

|a_m|+ 2bm−1

∆

am−1

bm−1

+bm−2

∆

an

bn

i

: 0< x <1 C2(m, n, r, x)

h

|am|+ 2bm−1

∆

am−1

bm−1

+b_m−2

∆

an

bn

i

(ii) If {a_k}is a nonnegative sequence such that{a_kb_k}is a convex sequence,

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then

m

X

k=n

a_kw_k(x)

≤











C₁(m, n, x) [|a_n|+ 2b⁻¹_n |∆(a_nb_n)|

+|∆(am−1bm−1)|] : 0< x <1 C2(m, n, r, x) [|an|+ 2b⁻¹_n |∆(anbn)|

+|∆(am−1bm−1)|] :x∈(2^−r,2^−r+1) for all n, m, r ∈N, m > n.

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References

[1] G. ALEXITS, Convergence problems of orthogonal series, Pergamon Press New York- Oxford-Paris, 1961.

[2] N.J. FINE, On the Walsh functions, Trans. Amer. Math. Soc., 65 (1949), 372–414.

[3] B.I. GOLUBOV, A.V. EFIMOVAND V.A. SKVORCOV, Walsh series and transformations, Science, Moscow (1987) (Russian).

[4] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer, 1993.

[5] D.S. MITRINOVI ´C AND J.E. PE ˇCARI ´C, On an inequality of G.K.Lebed, Prilozi MANU, Od. Mat. Tehn. Nauki, 12 (1991), 15–19.

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

Belgrade, 2 (1933), 45–59.

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

[8] Ž. TOMOVSKI, Some new inequalities for complex trigonometric polyno- mials with special coefficients, J. Inequal. in Pure and Appl. Math., 4(4) (2003), Art. 78. [ONLINE http://jipam.vu.edu.au/article.

php?sid=319].