Journal of Inequalities in Pure and
Applied Mathematics
NOTE ON BERNSTEIN’S INEQUALITY FOR THE THIRD
DERIVATIVE OF A POLYNOMIAL
volume 5, issue 1, article 7,
2004.
CLÉMENT FRAPPIER
Département de Mathématiques et de Génie Industriel
École Polytechnique de Montréal,
C.P. 6079, succ. Centre-ville
Montréal (Québec) H3C 3A7
CANADA.
EMail: clement.frappier@polymtl.ca
Received 05 November, 2003;
accepted 13 February, 2004.
Communicated by: R.N. Mohapatra
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
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Given a polynomial p(z) = nj=0 aj z j , we give the best possible constant c3 (n)
such that kp000 k + c3 (n)|a0 | ≤ n(n − 1)(n − 2)kpk, where k k is the maximum
norm on the unit circle {z : |z| = 1}. Most of the computations are done with a
computer.
2000 Mathematics Subject Classification: 26D05, 26D10, 30A10.
Key words: Bernstein’s inequality, Unit circle.
The author was supported by Natural Sciences and Engineering Research Council
of Canada Grant 0GP0009331.
Contents
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
Title Page
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Two Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
P
Let Pn denote the class of all polynomials p(z) = nj=0 aj z j , of degree ≤ n
with complex coefficients. The famous Bernstein’s inequality states that
(1.1)
kp0 k ≤ nkpk,
where kpk := max|z|=1 |p(z)|. The inequality (1.1) has been refined and generalized in numerous ways; see [3] for many interesting results. It is obvious
from (1.1) that
(1.2)
kp(k) k ≤
n!
kpk
(n − k)!
Clément Frappier
for 1 ≤ k ≤ n. Let ck (n) be the best possible constant such that
(1.3)
kp(k) k + ck (n)|a0 | ≤
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n!
kpk.
(n − k)!
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By “bestP
possible” we mean that, for every ε > 0, there exists a polynomial
pε (z) = nj=0 aj (ε)z j such that
kpε(k) k + (ck (n) + ε) |a0 (ε)| >
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
n!
kpε k.
(n − k)!
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2n
,
n+2
It is known (see [4, p. 125] or [2, p. 70]) that c1 (1) = 1 and c1 (n) =
n ≥ 2.
2(n−1)(2n−1)
We [1, p. 30] also have c2 (n) =
, n ≥ 1. The aim of this note is to
(n+1)
prove the following result.
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Theorem 1.1. Let p ∈ Pn , p(z) =
possible constant such that
Pn
j=0
aj z j . If we denote by c3 (n) the best
kp000 k + c3 (n)|a0 | ≤ n(n − 1)(n − 2)kpk
(1.4)
then c3 (1) = c3 (2) = 0 and, for n ≥ 3,
c3 (n) =
(1.5)
A(n)
,
B(n)
where
3
3
A(n) := 6(n − 2)(n − 1)
2
(n − 1) (8n − 15n + 6)
n(n − 2)
2
3
2
+ (n − 1) (2n + 7n − 21n + 6)Tn
(n − 1)2
n(n − 2)
3
2
− n(11n − 47n + 56n − 14)Un
(n − 1)2
and
B(n) := n 4(n − 1)5 (2n − 1)
n(n − 2)
+ 2(n − 1) (n + 3n − 13n + 10n − 2)Tn
(n − 1)2
n(n − 2)
4
3
2
− n(11n − 54n + 86n − 50n + 9)Un
.
(n − 1)2
4
3
Clément Frappier
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Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
2
sin((n+1) arccos(x))
sin(arccos(x))
Here, Tn (x) := cos (n arccos(x)) and Un (x) :=
respectively the Chebyshev polynomials of the first and second kind.
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2.
Proof of the Theorem
The method of proof used to obtain inequalities of the type (1.3) is well described in the aforementioned references. We give some details for the sake of
completeness. Consider two analytic functions
f (z) =
∞
X
j
aj z ,
g(z) =
j=0
∞
X
bj z j
j=0
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
for |z| ≤ K. The function
(f ? g)(z) :=
∞
X
aj b j z j
(|z| ≤ K)
Clément Frappier
j=0
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is said to be their Hadamard product.
Let Bn be the class of polynomials Q in Pn such that
kQ ? pk ≤ kpk
Contents
for every p ∈ Pn .
To p ∈ Pn we associate the polynomial pe(z) := z n p
1
z̄
. Observe that
e ∈ Bn .
Q ∈ Bn ⇐⇒ Q
Let us denote by Bn0 the subclass of Bn consisting of polynomials R in Bn for
which R(0) = 1. The following lemma contains a useful characterization of
Bn0 .
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P
Lemma 2.1. [2] The polynomial R(z) = nj=0 bj z j , where b0 = 1, belongs to
Bn0 if and only if the matrix


b0
b1 . . . bn−1 bn
 b̄
b0 . . . bn−2 bn−1 
 1

 .

.
.
.
.

..
..
..
.. 
M (b0 , b1 , . . . , bn ) :=  ..



b̄n−1 b̄n−2 . . . b0
b1 
b̄n b̄n−1 . . . b̄1
b0
is positive semi-definite.
The following well-known result enables us to study the definiteness of the
e
matrix
Pn Mj(1, b1 , . . . , bn ) associated with the polynomial R(z) = Q(z) = 1 +
j=1 bj z .
Lemma 2.2. The hermitian matrix


a11 a12 . . . a1n


 a21 a22 . . . a2n 


..
..
..  ,
 ..
.
.
. 
 .
an1 a12 . . . ann
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
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aij = āji ,
is positive definite if and only if its leading principal minors are all positive.
Here we simply use the calculations done in [1], where Lemmas 2.1 and 2.2
are applied to a polynomial of the form
n−1
X
α
j(j − 1)(j − 2) j
R(z) = 1 +
z +
zn.
n(n
−
1)(n
−
2)
n(n
−
1)(n
−
2)
j=1
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In that paper, the evaluation of the best possible constant c3 (n) is reduced to the
evaluation of the least positive root of the quadratic polynomial in c
−6(n−1)2
0
−c −6(n−1)2 12n(n−2)
c
∗
6+c x1,n−4 y1,n−4
6n(n−2)
−6−
(n−1)2
−c
c
x
y
−6(n−1)(n−2)
6(n−2)−
(2.1) D(n, c) := ,
2,n−4
2,n−4
(n−1)2 c
x3,n−5
y3,n−5
3(n−1)(n−2)
−3(n−1)(n−2) −c
x
y
(n−1)(n−2)(n−3)
n(n−1)(n−2)
4,n−6
4,n−6
where
xj,1 = Hj,1 +
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
12n(n−2)
,
(n−1)2
2n(n − 2)
xj,1 , yj,1 = Hj,1 − 6,
(n − 1)2
2n(n − 2)
xj,k−1 + xj,k−2 = Hj,k ,
xj,k −
(n − 1)2
y1,k + x1,k−1 = 6 for 2 ≤ k ≤ n − 5,
yj,k + xj,k−1 = Hj,k ,
∗
y1,n−4 + x1,n−5 = −6n(n − 2)
Clément Frappier
xj,2 = yj,1 +
for j = 1, 2, 3, 4, and


6




6(k + 1)
Hj,k =

3(k + 1)(k + 2)





(k + 1)(k + 2)(k + 3)
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if j = 1,
1 ≤ k ≤ n − 4,
if j = 2,
1 ≤ k ≤ n − 4,
if j = 3,
1 ≤ k ≤ n − 5,
if j = 4,
1 ≤ k ≤ n − 6.
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It was impossible at the time of [1] to obtain a simple expression for D(n, c).
With the development of mathematical software, it has become possible to handle nearly all the difficulties. The following computations can be done with
Mathematica 4.1.
The determinant D(n, c) can be expressed in the form
D(n, c) = q0 (n) − q1 (n)c − q2 (n)c2 ,
(2.2)
where
(2.3)
81(n − 2)(n − 1)8
q0 (n) :=
8(n − 1)(2n − 3)(2n2 − 4n + 1)
(2n2 − 4n + 1)
√
n
+ (n − 1)−2n n(n − 2) − i 2n2 − 4n + 1
2
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
2
× 2(2n − 4n + 1)(n + 2n − 6)
√
− i(n − 2)(11n2 − 20n + 3) 2n2 − 4n + 1
−2n
+ (n − 1)
n(n − 2)
√
n
+ i 2n2 − 4n + 1 2(2n2 − 4n + 1)(n2 + 2n − 6)
√
2
2
+ i(n − 2)(11n − 20n + 3) 2n − 4n + 1 ,
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5
(2.4)
54(n − 1)
(n − 1)(7n2 − 14n + 6)(2n2 − 4n + 1)
(2n2 − 4n + 1)
√
n
+ (n − 1)−2n n(n − 2) − i 2n2 − 4n + 1
q1 (n) :=
× (5n2 − 10n + 3) (2n2 − 4n + 1)
√
− in(n − 2) 2n2 − 4n + 1 + (n − 1)−2n n(n − 2)
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√
n
+ i 2n2 − 4n + 1 (5n2 − 10n + 3) (2n2 − 4n + 1)
√
2
+ in(n − 2) 2n − 4n + 1 ,
and
9n(n − 1)2
(2.5) q2 (n) :=
8(n − 1)3 (2n − 1) + (n − 1)−2n n(n − 2)
2
4(2n − 4n + 1)
√
n
− i 2n2 − 4n + 1 2(2n2 − 4n + 1)
× (n4 + 3n3 − 13n2 + 10n − 2)
√
− in(11n4 − 54n3 + 86n2 − 50n + 9) 2n2 − 4n + 1
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
−2n
+ (n − 1)
n(n − 2)
√
n
+ i 2n2 − 4n + 1 2(2n2 − 4n + 1)
× (n4 + 3n3 − 13n2 + 10n − 2)
√
4
3
2
2
+ in(11n − 54n + 86n − 50n + 9) 2n − 4n + 1 .
The only real problem we encountered was that the software was unable to
recognize that the discriminant q12 + 4q0 q2 is a perfect square. It is necessary to
observe that
(2.6) q12 + 4q0 q2
√
27(n − 1)−2n+9
= √
4(2n − 1)(2n − 3)(n − 1)2(n−1) 2n2 − 4n + 1
2n2 − 4n + 1
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√
+ 2n(n − 2) 2n2 − 4n + 1 − i(n − 1)(11n2 − 22n + 6) n(n − 2)
√
√
n−1
− i 2n2 − 4n + 1
+ 2n(n − 2) 2n2 − 4n + 1
√
n−1 2
2
+ i(n − 1)(11n − 22n + 6) n(n − 2) + i 2n2 − 4n + 1
.
The positive root of D(n, c) is readily found with the√help of (2.6). That root
2n2 −4n+1
can be written in the form (1.5) where eit := n(n−2)−i
. Throughout
(n−1)2
the calculations, the identity
n n
√
√
2
2
n(n − 2) − i 2n − 4n + 1
n(n − 2) + i 2n − 4n + 1 = (n − 1)4n
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
is useful for simplifying the expressions.
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3.
Two Open Questions
We immediately obtain the values c3 (3) = 6, c3 (4) = 156
, c3 (5) = 5736
, c3 (6) =
7
115
92955
342430
, c3 (7) = 2443 , etc. It is highly probable that all the constants ck (n),
1043
appearing in (1.3), are rational numbers. Other values are c4 (4) = 24, c4 (5) =
2184
, c4 (6) = 11808
, c4 (7) = 11625
, etc. In fact, all the constants ck,ν (n), related
19
37
17
to the same kind of inequality with |aν | rather than |a0 |, seem to be rational
numbers.
Question 1. Are the constants ck,ν (n) rational numbers?
As far as the constants ck (n), k ≥ 4, are concerned, it is probable that a few
simple expressions can be found for them. The most interesting problem here
is perhaps to find the asymptotic value of ck (n) as n → ∞. We have
(3.1)
ck (n)
= 2k
n→∞ nk−1
lim
for k = 0, 1, 2 and k = 3. The latter case follows from (1.5).
Question 2. Does (3.1) hold for k ≥ 4?
Note On Bernstein’s Inequality
For The Third Derivative Of A
Polynomial
Clément Frappier
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References
[1] C. FRAPPIER AND M.A. QAZI, A refinement of Bernstein’s inequality for
the second derivative of a polynomial, Ann. Univ. Mariae Curie-Sklodowska
Sect. A, LII(2,3) (1998), 29–36.
[2] C. FRAPPIER, Q.I. RAHMAN AND St. RUSCHEWEYH, New inequalities
for polynomials, Trans. Amer. Math. Soc., 288(1) (1985), 69–99.
[3] Q.I. RAHMAN AND G. SCHMEISSER, Analytic Theory of Polynomials,
Oxford Science Publications, Oxford 2002.
[4] St. RUSCHEWEYH, Convolutions in Geometric Function Theory, Les
Presses de l’Université de Montréal, Montréal, 1982.
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For The Third Derivative Of A
Polynomial
Clément Frappier
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