Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES FOR WALSH POLYNOMIALS WITH SEMI-MONOTONE
AND SEMI-CONVEX COEFFICIENTS
volume 6, issue 4, article 98,
2005.
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
Received 08 April, 2005;
accepted 03 September, 2005.
Communicated by: D. Stefanescu
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
112-05
Quit
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 coefficients, Complex convex coefficients, Complex semi-monotone coefficients, Complex monotone coefficients, Fine inequality.
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Contents
1
Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Živorad Tomovski
3
6
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
1.
Introduction and Preliminaries
We consider the Walsh orthonormal system {wn (x)}∞
n=0 defined on [0, 1) in the
Paley enumeration. Thus w0 (x) ≡ 1 and for each positive integer with dyadic
development
p
X
n=
2νi , ν1 > ν2 > · · · > νp ≥ 0,
i=1
we have
wn (x) =
p
Y
rνi (x),
i=1
{rn (x)}∞
n=0
where
denotes the Rademacher system of functions defined by (see,
e.g. [1, p. 60], [3, p. 9-10])
rν (x) = sign sin 2ν π(x) (ν = 0, 1, 2, . . . ; 0 ≤ x < 1).
P
In this paper we shall consider the Walsh polynomials m
k=n λk wk (x) with
complex-valued coefficients {λk }.
Let ∆λn = λn − λn+1 and ∆2 λn = ∆(∆λn ) = ∆λn − ∆λn+1 = λn −
2λn+1 + λn+2 , for all n = 1, 2, 3 . . . .
Petrović [6] proved the following complementary triangle inequality for a
sequence 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
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 15
ν=1
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
For 0 < θ < π2 denote by K(θ) the cone K(θ) = {z : | arg z| ≤ θ}.
Let {bk } be a positive nondecreasing sequence. The following definitions
are given in [7] and [8]. The sequence of complex numbers {uk } issaid to
be complex semi−monotone if there exists a cone K(θ) such that ∆ ubkk ∈
K(θ) or ∆(uk bk ) ∈ K(θ). For bk = 1, the sequence {uk } 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 cone K(θ), such that ∆2 ubkk ∈
K(θ) or ∆2 (uk bk ) ∈ 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].
P
Theorem B ([7]). Let {zk } be a sequence such that | m
k=n zk | ≤ A, (∀n, m ∈
N, m > n), where A is a positive number.
(i) If ∆ ubkk ∈ K(θ), then
m
X
1
1 bm
uk zk ≤ A 1 +
|um | +
|un | , (∀n, m ∈ N, m > n).
cos θ
cos θ bn
k=n
(ii) If ∆(uk bk ) ∈ K(θ), then
m
X
1
1 bm
uk zk ≤ A 1 +
|un | +
|um | ,
cos θ
cos θ bn
k=n
P
q P
j
Theorem C ([8]). Let A = max zk .
n≤p≤q≤m j=p k=i
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
(∀n, m ∈ N, m > n).
Close
Quit
Page 4 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
(i) If {uk } is a sequence of complex numbers such that ∆2
uk
bk
∈ K(θ), then
m
X
1
u
b
m−1
m
∆
+
∆ un ,
uk zk ≤ A |um | + bm 1 +
cos θ
bm−1
cos θ
bn k=n
(∀n, m ∈ N, m > n).
(ii) 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
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
(∀n, m ∈ N, m > n).
Title Page
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.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
2.
Main Results
For the main results we require the following Lemma.
Lemma 2.1. For all p, q, r ∈ N, p < q the following inequalities hold:
q
X
2
(i) ωk (x) ≤ , 0 < x < 1.
x
k=p
q j
X X
(ii) ωk (x)
j=p k=l
 2(q−p+1)
 x
= C1 (p, q, x) : 0 < x < 1
≤
 8
+ x82 + 2(q−p+1)
+ 1 = C2 (p, q, r, x) : x ∈ (2−r , 2−r+1 )
x(x−2−r )
x
Pq−1
Proof. (i) Let Dq (x) = i=0
wi (x) be the Dirichlet kernel. Then it is known
that (see [3, p. 28]) |Dq (x)| ≤ x1 , 0 < x < 1. Hence
q
X
2
wk (x) = |Dq+1 (x) − Dp (x)| ≤ |Dq+1 (x)| + |Dp (x)| ≤ .
x
k=p
(ii) By (i) we get
q j
q j
X X
X
2(q − p + 1)
X
wk (x) ≤
wk (x) ≤
, 0 < x < 1.
x
j=p k=l
j=p k=l
Pn
1
Let Fn (x) = n+1
k=0 Dk (x) be the Fejer kernel. Applying Fine’s inequality
(see [2])
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
(n + 1)Fn (x) <
4
4
+
,
x(x − 2−r ) x2
x ∈ (2−r , 2−r+1 ),
we get
q j
q
X X
X
wk (x) = (Dj+1 (x) − Dl (x))
j=p k=l
j=p
q
X
q−p+1
≤
Dj+1 (x) +
x
j=p
≤ |(q + 1)Fq (x)| + |Dq+1 (x)| + |D0 (x)|
q−p+1
+ |pFp−1 (x)| +
x
2(q − p + 1)
8
8
<
+
+
+ 1,
x(x − 2−r ) x2
x
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
x ∈ (2−r , 2−r+1 ).
Title Page
Contents
Applying the inequality (i) of the above lemma and Theorem B, we obtain
following theorem.
Theorem 2.2. Let 0 < x < 1.
(i) If {uk } is a sequence of complex numbers such that ∆
uk
bk
∈ K(θ), then
m
X
2 1
1 bm
1+
|um | +
|un | ,
uk wk (x) ≤
x
cos θ
cos θ bn
k=n
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 15
(∀n, m ∈ N, m > n).
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
(ii) If {uk } is a sequence of complex numbers such that ∆(uk bk ) ∈ K(θ), then
m
X
2 1
1 bm
uk wk (x) ≤
1+
|un | +
|um | ,
x
cos
θ
cos
θ
b
n
k=n
(∀n, m ∈ N, m > n).
Specially for bk = 1 we get the following inequalities for Walsh polynomials
with complex monotone coefficients.
Corollary 2.3. Let 0 < x < 1. If {uk } is a sequence of complex numbers such
that ∆uk ∈ K(θ), then
m
X
2 1
1
uk ωk (x) ≤
1+
|um | +
|un | , (∀n, m ∈ N, m > n).
x
cos
θ
cos
θ
k=n
Corollary 2.4. Let 0 < x < 1. If {uk } is a complex monotone sequence such
that lim uk = 0, then
k→∞
∞
X
2
uk ωk (x) ≤
|un |.
x cos θ
k=n
In [4] (chapter XXI), [5] Mitrinović and Pečarić 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.
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
Corollary 2.5. Let 0 < x < 1.
(i) If {ak } is a nonnegative sequence such that {ak b−1
k } is a decreasing sequence, then
m
X
a b m
n
ak wk (x) ≤
, (∀n, m ∈ N, m > n).
x bn
k=n
(ii) If {ak } is a nonnegative sequence such that {ak bk } is an increasing sequence, then
m
X
a b m
m
ak wk (x) ≤
, (∀n, m ∈ N, m > n).
x
bn
k=n
Now, applying the inequality (ii) of Lemma 2.1, we obtain new inequalities
for Walsh polynomials with complex semi-convex coefficients.
Theorem 2.6.
(i) If {uk } is a sequence of complex numbers such that ∆2 ubkk ∈ K(θ), then
h

um−1 1

C
(m,
n,
x)
|u
|
+
b
1
+
∆ bm−1  1
m
m−1
cos θ 

i



m−2 + bcos
∆ ubnn : 0 < x < 1
m
X


θ uk wk (x) ≤
h
um−1 
1

k=n

C
(m,
n,
r,
x)
|u
|
+
b
1
+
∆ bm−1 2
m
m−1

cos θ 

i


m−2 
+ bcos
∆ ubnn : x ∈ (2−r , 2−r+1 )
θ for all n, m, r ∈ N, m > n.
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
(ii) If {uk } is a sequence of complex numbers such that ∆2 (uk bk ) ∈ K(θ),
then

1
−1

C1 (m, n, x) |un | + bn 1 + cos θ




m
×(|∆(un bn )| + |∆(um−1 bm−1 )|)] : 0 < x < 1


X
uk wk (x) ≤


k=n

C2 (m, n, r, x) |un | + b−1
1 + cos1 θ
n





×(|∆(un bn )| + |∆(um−1 bm−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
m−1 j
m
u X
X u
um−1 X X
m
k
(bk wk ) = bk wk + ∆
bk wk
bm
bk
bm−1 j=n k=n
k=n
k=n
X
j
m−2
r X
X
u
r
+
∆2
bk wk b
r
r=n
j=n k=n
m−1
j
m
X X
|um | X u
m−1
≤
bm wk + bm−1 ∆
wk bm
bm−1 j=n k=n k=n
j
r X
m−2
X ur X
∆2
w
+ bm−2
k .
br
r=n
j=n k=n
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
Using the Petrović inequality and inequality (ii) of Lemma 2.1, we obtain:
m−1
j
m
m
X
X
X
X
u
m−1 u
w
(x)
w
≤
|u
|
w
+
b
∆
k k
k
m k
m−1 b
m−1
j=n k=n
k=n
k=n
r j
m−2
bm−2 X 2 ur X X +
∆
wk cos θ r=n
br j=n k=n h

um−1 1

C1 (m, n, x) |um | + bm−1 1 + cos θ ∆ bm−1 


i


bm−2 

+ cos θ ∆ ubnn : 0 < x < 1

≤
h
um−1 
1


C2 (m, n, r, x) |um | + bm−1 1 + cos θ ∆ bm−1 


i


bm−2 
+ cos θ ∆ ubnn : x ∈ (2−r , 2−r+1 )
m
X
−1
(uk bk )bk wk k=n
j
m−1
m
m
X
X
X
X
−1
2
= un bn
bk wk −
∆ (uj−1 bj−1 )
b−1
k wk
r=n
j=n+1
k=r
m X
m
m
X
X
−1
w
−
∆(u
b
)
w
+∆(un bn )
b−1
b
k
m−1 m−1
k
k
k
r=n
k=n
Živorad Tomovski
Title Page
Contents
(ii) Analogously as the proof of (i), we obtain:
k=n
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 15
k=r
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
j
m−1
m
m
X X
X
X −1
2
w
w
+
b
∆
(u
b
)
≤ |un |bn b−1
k
j−1
j−1
k
n
n
r=n
j=n+1
k=n
k=r
m
m
m X
X
X
−1
+ b−1
|∆(u
b
)|
w
+
b
|∆(u
b
)|
w
n n k
m−1 m−1 k .
n
n
r=n
k=n
k=r
Hence,
m
m
j
m
X
X −1 m−1
X
X
X
bn 2
∆ (uj−1 bj−1 )
wk uk wk (x) ≤ |un | wk +
cos θ r=n k=r
j=n+1
k=n
k=n
m
m m
X X X + b−1
wk + b−1
wk n |∆(un bn )| n |∆(um−1 bm−1 )| r=n k=r
k=n


C1 (m, n, x) |un | + b−1
1 + cos1 θ

n





×(|∆(un bn )| + |∆(um−1 bm−1 )|)] : 0 < x < 1,

≤



C2 (m, n, r, x) |un | + b−1
1 + cos1 θ
n





×(|∆(u b )| + |∆(u
b
)|)] : x ∈ (2−r , 2−r+1 ).
n n
m−1 m−1
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
If bk = 1, k = n, n + 1, . . . , m from Theorem 2.6, we obtain the following
corollary.
Quit
Page 12 of 15
Corollary 2.7. Let {uk } be a complex-convex sequence. Then,
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au

1
C
(m,
n,
x)
|u
|
+
1
+

1
m
cos θ

m


1
X


×|∆u
|
+
|∆u
|
:0<x<1
m−1
n
cos θ
uk wk (x) ≤


C2 (m, n, r, x) |um | + 1 + cos1 θ
k=n




×|∆um−1 | + cos1 θ |∆un | : x ∈ (2−r , 2−r+1 )
for all n, m, r ∈ N, m > n.
Remark 1. Similarly, the results of Theorem 2.2, Theorem 2.6, Corollary 2.3,
Corollary 2.5 and Corollary 2.7 were given by the author in [7, 8] for trigonometric polynomials with complex valued coefficients.
Corollary 2.8.
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
{ak b−1
k }
(i) If {ak } is a nonnegative sequence such that
is a convex sequence,
then

h
am−1 
C (m, n, x) |am | + 2bm−1 ∆ bm−1 

 1
i


m
X


+bm−2 ∆ abnn : 0 < x < 1
ak wk (x) ≤
h

C (m, n, r, x) |a | + 2b
am−1 ∆
k=n

2
m
m−1
bm−1 
i




+bm−2 ∆ abnn : x ∈ (2−r , 2−r+1 )
for all n, m, r ∈ N, m > n.
(ii) If {ak } is a nonnegative sequence such that {ak bk } is a convex sequence,
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
then


C1 (m, n, x) [|an | + 2b−1

n |∆(an bn )|





+|∆(am−1 bm−1 )|] : 0 < x < 1

m
X
ak wk (x) ≤


k=n

C2 (m, n, r, x) [|an | + 2b−1
n |∆(an bn )|





+|∆(am−1 bm−1 )|] : x ∈ (2−r , 2−r+1 )
for all n, m, r ∈ N, m > n.
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au
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. EFIMOV AND V.A. SKVORCOV, Walsh series and
transformations, Science, Moscow (1987) (Russian).
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer, 1993.
[5] D.S. MITRINOVIĆ AND J.E. PEČARIĆ, On an inequality of G.K.Lebed,
Prilozi MANU, Od. Mat. Tehn. Nauki, 12 (1991), 15–19.
[6] M. PETROVIĆ, Theoreme sur les integrales curvilignes, Publ. Math. Univ.
Belgrade, 2 (1933), 45–59.
[7] Ž. TOMOVSKI, On some inequalities of Mitrinovic and Pečarić, Prilozi
MANU, Od. Mat. Tehn. Nauki, 22 (2001), 21–28
[8] Ž. TOMOVSKI, Some new inequalities for complex trigonometric polynomials 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].
Inequalities for Walsh
Polynomials with
Semi-Monotone and
Semi-Convex Coefficients
Živorad Tomovski
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 15
J. Ineq. Pure and Appl. Math. 6(4) Art. 98, 2005
http://jipam.vu.edu.au