Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 98, 2005
INEQUALITIES FOR WALSH POLYNOMIALS WITH SEMI-MONOTONE AND
SEMI-CONVEX COEFFICIENTS
ŽIVORAD TOMOVSKI
U NIVERSITY "S T. C YRIL AND M ETHODIUS "
FACULTY OF NATURAL S CIENCES AND M ATHEMATICS
I NSTITUTE OF M ATHEMATICS
PO B OX 162, 91000 S KOPJE
R EPUBIC OF M ACEDONIA
tomovski@iunona.pmf.ukim.edu.mk
Received 08 April, 2005; accepted 03 September, 2005
Communicated by D. Stefanescu
A BSTRACT. 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.
Key words and phrases: Petrovic inequality, Walsh polynomial, Complex semi-convex coefficients, Complex convex coefficients, Complex semi-monotone coefficients, Complex monotone coefficients, Fine inequality.
2000 Mathematics Subject Classification. 26D05, 42C10.
1. I NTRODUCTION AND P RELIMINARIES
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
n=
p
X
2νi ,
ν1 > ν2 > · · · > νp ≥ 0,
i=1
we have
wn (x) =
p
Y
rνi (x),
i=1
where {rn (x)}∞
n=0 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 }.
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
112-05
2
Ž IVORAD T OMOVSKI
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
zν ≥ (cos θ)
|zν |.
ν=1
ν=1
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 } is said 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
2 uk
said to be complex semi−convex if there exists a cone K(θ), such that ∆ bk ∈ 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
b
n
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
(∀n, m ∈ N, m > n).
(i) If {uk } is a sequence of complex numbers such that ∆ ubkk ∈ 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
2
(∀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
(∀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.
J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
http://jipam.vu.edu.au/
I NEQUALITIES FOR WALSH P OLYNOMIALS
3
2. M AIN R ESULTS
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
ωk (x) ≤ , 0 < x < 1.
(i) x
k=p
q j
 2(q−p+1)
X X
 x
= C1 (p, q, x) : 0 < x < 1
(ii) ωk (x) ≤
 8
+ 8 + 2(q−p+1) + 1 = C (p, q, r, x) : x ∈ (2−r , 2−r+1 )
j=p k=l
x(x−2−r )
x2
x
2
P
Proof. (i) Let Dq (x) = q−1
i=0 wi (x) be the Dirichlet kernel. Then it is known that (see [3, p.
1
28]) |Dq (x)| ≤ x , 0 < x < 1. Hence
q
X
2
w
(x)
= |Dq+1 (x) − Dp (x)| ≤ |Dq+1 (x)| + |Dp (x)| ≤ .
k
x
k=p
(ii) By (i) we get
q j
q j
X X
X
2(q − p + 1)
X
wk (x) ≤
wk (x) ≤
,
x
j=p k=l
j=p k=l
Let Fn (x) =
1
n+1
Pn
k=0
0 < x < 1.
Dk (x) be the Fejer kernel. Applying Fine’s inequality (see [2])
(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)| + |pFp−1 (x)| +
<
8
8
2(q − p + 1)
+ 2+
+ 1,
−r
x(x − 2 ) x
x
q−p+1
x
x ∈ (2−r , 2−r+1 ).
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 ∆ ubkk ∈ K(θ), then
m
X
2 1
1 bm
uk wk (x) ≤
1+
|um | +
|un | , (∀n, m ∈ N, m > n).
x
cos θ
cos θ bn
k=n
J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
http://jipam.vu.edu.au/
4
Ž IVORAD T OMOVSKI
(ii) If {uk } is a sequence of complex numbers such that ∆(uk bk ) ∈ K(θ), then
m
2 X
1
1 bm
uk wk (x) ≤
1+
|un | +
|um | , (∀n, m ∈ N, m > n).
x
cos θ
cos θ bn
k=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,
k→∞
then
∞
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.
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 n
m
ak wk (x) ≤
, (∀n, m ∈ N, m > n).
x
b
n
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
uk
bk
∈ K(θ), then
h

um−1 1

C
(m,
n,
x)
|u
|
+
b
1
+
∆ bm−1 
1
m
m−1
cos θ 

i

bm−2 

un 
∆
:0<x<1
+
m
X

cos θ
bn uk wk (x) ≤
h
um−1 
1

k=n

C2 (m, n, r, x) |um | + bm−1 1 + cos θ ∆ bm−1 


i


m−2 
+ bcos
∆ ubnn : x ∈ (2−r , 2−r+1 )
θ for all n, m, r ∈ N, m > n.
J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
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I NEQUALITIES FOR WALSH P OLYNOMIALS
5
(ii) If {uk } is a sequence of complex numbers such that ∆2 (uk bk ) ∈ K(θ), then

1
−1
C
(m,
n,
x)
|u
|
+
b
1
+

1
n
n
cos
θ





×(|∆(un bn )| + |∆(um−1 bm−1 )|)] : 0 < x < 1

m
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−1 j
m
m
X
u X
um−1 X X
uk
m
(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
ur
2
+
∆
bk wk br j=n k=n
r=n
m
m−1
j
X
X
|um | X u
m−1 ≤
bm wk + bm−1 ∆
w
k
bm
b
m−1
j=n k=n
k=n
r j
m−2
X ur X
X ∆2
+ bm−2
wk .
br r=n
j=n k=n
Using the Petrović inequality and inequality (ii) of Lemma 2.1, we obtain:
m−1
j
m
m
X
X
X
X
um−1 uk wk (x) ≤ |um | wk + bm−1 ∆
wk bm−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

∆ bm−1 C
(m,
n,
x)
|u
|
+
b
1
+

1
m
m−1
cos θ 

i



m−2 
+ bcos
∆ ubnn : 0 < x < 1

θ ≤
h
um−1 
1


C2 (m, n, r, x) |um | + bm−1 1 + cos θ ∆ bm−1 


i


m−2 
+ bcos
∆ ubnn : x ∈ (2−r , 2−r+1 )
θ (ii) Analogously as the proof of (i), we obtain:
J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
http://jipam.vu.edu.au/
6
Ž IVORAD T OMOVSKI
j
m−1
m
m
m
X
X
X
X
X
−1
−1
2
b−1
bk wk −
∆ (uj−1 bj−1 )
(uk bk )bk wk = un bn
k wk
r=n
j=n+1
k=n
k=n
+∆(un bn )
m
X
b−1
k wk − ∆(um−1 bm−1 )
k=r
m X
m
X
r=n k=r
k=n
−1
bk wk j
m
m−1
m
X
X X
X −1
2
w
≤ |un |bn b−1
+
b
∆
(u
b
)
w
k
j−1
j−1
k
n
n
r=n
j=n+1
k=n
k=r
m
m
m X
X
X
−1
−1
+ bn |∆(un bn )| wk + bn |∆(um−1 bm−1 )| wk .
r=n k=r
k=n
Hence,
m
m
j
m
X
X −1 m−1
X
X
X
b
∆2 (uj−1 bj−1 )
wk uk wk (x) ≤ |un | wk + n cos θ r=n k=r
j=n+1
k=n
k=n
m
m X
m
X
X
−1
−1
+ bn |∆(un bn )| wk + bn |∆(um−1 bm−1 )| wk 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,

≤


1 + cos1 θ
C2 (m, n, r, x) |un | + b−1
n




×(|∆(un bn )| + |∆(um−1 bm−1 )|)] : x ∈ (2−r , 2−r+1 ).
If bk = 1, k = n, n + 1, . . . , m from Theorem 2.6, we obtain the following corollary.
Corollary 2.7. Let {uk } be a complex-convex sequence. Then,

1
C
(m,
n,
x)
|u
|
+
1
+

1
m
cos θ




1

×|∆um−1 | + cos θ |∆un | : 0 < x < 1

m
X
uk wk (x) ≤

1

k=n
C
(m,
n,
r,
x)
|u
|
+
1
+

2
m

cos θ



1
×|∆um−1 | + cos θ |∆un | : x ∈ (2−r , 2−r+1 )
for all n, m, r ∈ N, m > n.
Remark 2.8. 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.
J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
http://jipam.vu.edu.au/
I NEQUALITIES FOR WALSH P OLYNOMIALS
7
Corollary 2.9.
(i) If {ak } is a nonnegative sequence such that {ak b−1
} is a convex sequence, then
k
h

am−1 
C1 (m, n, x) |am | + 2bm−1 ∆ bm−1 


i




+bm−2 ∆ abnn : 0 < x < 1
m
X

ak wk (x) ≤
h

am−1 
k=n

C2 (m, n, r, x) |am | + 2bm−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, then

C1 (m, n, x) [|an | + 2b−1

n |∆(an bn )|




m
+|∆(am−1 bm−1 )|] : 0 < x < 1
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.
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Moscow (1987) (Russian).
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Kluwer, 1993.
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A.M. FINK, Classical and New Inequalities in Analysis,
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Tehn. Nauki, 12 (1991), 15–19.
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J. Inequal. Pure and Appl. Math., 6(4) Art. 98, 2005
http://jipam.vu.edu.au/