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Transcript of Bernstein Beamer
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Bernstein Inequalities for Polynomials
Robert “Dr. Bob” Gardner
March 8, 2013
(Prepared in Beamer!)
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 1 / 35
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20th Anniversary
May 8, 1993
In 1993, I was desperate to get out of Louisiana and find a universityposition in a mountainous region of the country. On May 8, 1993 Iinterviewed here at ETSU for an assistant professor position. I gave apresentation on “Bernstein Inequalities for Polynomials and Other EntireFunctions.” This talk on the 20th anniversary of my interview reviews the
polynomial results and gives some updates on my work here at ETSU.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 2 / 35
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Introduction
The Complex Plane
The field of complex numbers consists of ordered pairs of real numbers,C = {(a, b ) | a, b ∈ R}, with addition defined as(a, b ) + (c , d ) = (a + c , b + d ) and multiplication defined as(a, b ) · (c , d ) = (ac − bd , bc + ad ). We denote (a, b ) as a + ib . Themodulus of z = a + ib is |z | =
√ a2 + b 2.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 3 / 35
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Introduction
Analytic Functions
Definition
A function f : G → C, where G is an open connected subset of C, isanalytic if f is continuously differentiable on G .
Theorem
If f is analytic in |z − a| < R then f (z ) =∞n=0 an(z − a)n for |z − a| < R, where the radius of convergence is at least R.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 4 / 35
I d i
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Introduction
Maximum Modulus Theorem.
Theorem
Let G be a bounded open set in C and suppose f is continuous on the closure of G, cl (G ), and analytic in G. Then
max{|f (z )| | z ∈ cl (G )} = max{|f (z )| | z ∈ ∂ G }.
In addition, if max{|f (z )| | z ∈ G } = max{|f (z )| | z ∈ ∂ G }, then f is constant.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 5 / 35
I t d ti
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Introduction
The Maximum Modulus Theorem for Unbounded Domains.
TheoremLet D be an open disk in the complex plane. Suppose f is analytic on the complement of D, continuous on the boundary of D, |f (z )| ≤ M on ∂ D,and lim
|z |→∞f (z ) = L for some complex L. Then |f (z )| ≤ M on the
complement of D.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 6 / 35
Introduction
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Introduction
Lucas’ Theorem.
Theorem
If all the zeros of a polynomial P lie in a half plane in the complex plane,then all the zeros of the derivative P lie in the same half plane.
Corollary
The convex polygon in the complex plane which contains all the zeros of apolynomial P, also contains all the zeros of P .
Fancois Lucas (1842–1891)Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 7 / 35
Introduction
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Introduction
Lucas’ Theorem.
Theorem
If all the zeros of a polynomial P lie in a half plane in the complex plane,then all the zeros of the derivative P lie in the same half plane.
Corollary
The convex polygon in the complex plane which contains all the zeros of apolynomial P, also contains all the zeros of P .
Fancois Lucas (1842–1891)Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 7 / 35
Motivation
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Motivation
Mendeleev’s Data
Russian chemist Dmitri Mendeleev studied the specific gravity of asolution as a function of the percentage of dissolved substance. His datacould be closely approximated by quadratic arcs and he wondered if thecorners where the arcs joined were real, or due to errors of measurement.His question, after normalization is: “If p (x ) is a quadratic polynomial
with real coefficients and |p (x )| ≤ 1 on [−1, 1], then how large can |p
(x )|be on [−1, 1]?”
A graph similar to Mendeleev’s Dmitri Mendeleev (1834–1907)
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 8 / 35
Motivation
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Motivation
Markov’s Result
Mendeleev answered his own question and showed that|p (x )
| ≤4 (and
the corners were determined to be genuine). Mendeleev told A. A. Markovabout his result, and Markov went on to prove the following.
Theorem
If p (x ) is a real polynomial of degree n, and |p (x )
| ≤1 on [
−1, 1] then
|p (x )| ≤ n2 on [−1, 1]. Equality holds only at ±1 and only whenp (x ) = ±T n(x ), where T n(x ) is the Chebyshev polynomial cos(n cos−1 x ).
Andrei Andreyevich Markov(1856–1922)
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 9 / 35
Motivation
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Markov’s Result
Mendeleev answered his own question and showed that|p (x )
| ≤4 (and
the corners were determined to be genuine). Mendeleev told A. A. Markovabout his result, and Markov went on to prove the following.
Theorem
If p (x ) is a real polynomial of degree n, and |p (x )
| ≤1 on [
−1, 1] then
|p (x )| ≤ n2 on [−1, 1]. Equality holds only at ±1 and only whenp (x ) = ±T n(x ), where T n(x ) is the Chebyshev polynomial cos(n cos−1 x ).
Andrei Andreyevich Markov(1856–1922)
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 9 / 35
Bernstein’s Results
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Such Inequalities in C
We want to address similar inequalities for polynomials over the complexfield.
So we are interested in comparing the size of polynomial P over |z | ≤ 1 tothe size of P over |z | ≤ 1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 10 / 35
Bernstein’s Results
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Such Inequalities in C
We want to address similar inequalities for polynomials over the complexfield.
So we are interested in comparing the size of polynomial P over |z | ≤ 1 tothe size of P over |z | ≤ 1.
That is, how large is max|z |≤1
|P (z )| in terms of max|z |≤1
|P (z )|?
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 10 / 35
Bernstein’s Results
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Such Inequalities in C
We want to address similar inequalities for polynomials over the complexfield.
So we are interested in comparing the size of polynomial P over |z | ≤ 1 tothe size of P over |z | ≤ 1.
That is, how large is max|z |≤1
|P (z )| in terms of max|z |≤1
|P (z )|?
In this direction, we prove the following lemma.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 10 / 35
Bernstein’s Results
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Such Inequalities in C
We want to address similar inequalities for polynomials over the complexfield.
So we are interested in comparing the size of polynomial P over |z | ≤ 1 tothe size of P over |z | ≤ 1.
That is, how large is max|z |≤1
|P (z )| in terms of max|z |≤1
|P (z )|?
In this direction, we prove the following lemma.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 10 / 35
Bernstein’s Results
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Bernstein’s Lemma.
Lemma
Let P and Q be polynomials such that (i) deg (P ) ≤ deg (Q ), (ii)|P (z )| ≤ |Q (z )| for |z | ≤ 1, and (iii) all zeros of Q lie in |z | ≤ 1. Then
|P (z )| ≤ |Q (z )| for |z | = 1.
Proof. Define f (z ) = P (z )/Q (z ). Then f is analytic on |z | > 1 and|f (z )| ≤ 1 for |z | = 1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 11 / 35
Bernstein’s Results
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Bernstein’s Lemma.
Lemma
Let P and Q be polynomials such that (i) deg (P ) ≤ deg (Q ), (ii)|P (z )| ≤ |Q (z )| for |z | ≤ 1, and (iii) all zeros of Q lie in |z | ≤ 1. Then
|P (z )| ≤ |Q (z )| for |z | = 1.
Proof. Define f (z ) = P (z )/Q (z ). Then f is analytic on |z | > 1 and|f (z )| ≤ 1 for |z | = 1.Next, if deg(P ) = deg(Q ) = n then
lim|z |→∞
f (z ) = L where L = an/b n
where an is the coefficient of z n in P and b n is the coefficient of z n in Q .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 11 / 35
Bernstein’s Results
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Bernstein’s Lemma.
Lemma
Let P and Q be polynomials such that (i) deg (P ) ≤ deg (Q ), (ii)|P (z )| ≤ |Q (z )| for |z | ≤ 1, and (iii) all zeros of Q lie in |z | ≤ 1. Then
|P (z )| ≤ |Q (z )| for |z | = 1.
Proof. Define f (z ) = P (z )/Q (z ). Then f is analytic on |z | > 1 and|f (z )| ≤ 1 for |z | = 1.Next, if deg(P ) = deg(Q ) = n then
lim|z |→∞
f (z ) = L where L = an/b n
where an is the coefficient of z n in P and b n is the coefficient of z n in Q .If deg(P ) < deg(Q ), then
lim|z |→∞
f (z ) = 0.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 11 / 35
Bernstein’s Results
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Bernstein’s Lemma.
Lemma
Let P and Q be polynomials such that (i) deg (P ) ≤ deg (Q ), (ii)|P (z )| ≤ |Q (z )| for |z | ≤ 1, and (iii) all zeros of Q lie in |z | ≤ 1. Then
|P (z )| ≤ |Q (z )| for |z | = 1.
Proof. Define f (z ) = P (z )/Q (z ). Then f is analytic on |z | > 1 and|f (z )| ≤ 1 for |z | = 1.Next, if deg(P ) = deg(Q ) = n then
lim|z |→∞
f (z ) = L where L = an/b n
where an is the coefficient of z n in P and b n is the coefficient of z n in Q .If deg(P ) < deg(Q ), then
lim|z |→∞
f (z ) = 0.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 11 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.Therefore |f (z 0)| = |P (z 0)/Q (z 0)| > 1 and so |z 0| < 1 by (∗).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.Therefore |f (z 0)| = |P (z 0)/Q (z 0)| > 1 and so |z 0| < 1 by (∗). Now if Q (z 0) = 0, then |z 0| ≤ 1. So all zeros of g lie in |z | ≤ 1. So by Lucas’Theorem, g has all its zeros in
|z
| ≤1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.Therefore |f (z 0)| = |P (z 0)/Q (z 0)| > 1 and so |z 0| < 1 by (∗). Now if Q (z 0) = 0, then |z 0| ≤ 1. So all zeros of g lie in |z | ≤ 1. So by Lucas’Theorem, g has all its zeros in
|z
| ≤1. So for no
|λ
|> 1 is
g (z ) = P (z ) − λQ (z ) = 0 where |z | > 1;
or in other words, P (z )/Q (z ) = λ where |λ| > 1 has no solution in|z | > 1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.Therefore |f (z 0)| = |P (z 0)/Q (z 0)| > 1 and so |z 0| < 1 by (∗). Now if Q (z 0) = 0, then |z 0| ≤ 1. So all zeros of g lie in |z | ≤ 1. So by Lucas’Theorem, g has all its zeros in |z | ≤ 1. So for no |λ| > 1 is
g (z ) = P (z ) − λQ (z ) = 0 where |z | > 1;
or in other words, P (z )/Q (z ) = λ where |λ| > 1 has no solution in|z | > 1. Hence |P (z )| ≤ |Q (z )| for |z | > 1. By taking limits, we have
|P
(z )| ≤ |Q
(z )| for |z | ≥ 1, and the result follows.Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein’s Lemma (continued).
So by the Maximum Modulus Theorem for Unbounded Domains,
|f (z )| ≤ 1 for |z | ≥ 1. (∗)
Let |λ| > 1 and define polynomial g (z ) = P (z ) − λQ (z ). If g (z 0) = P (z 0) − λQ (z 0) = 0 and if Q (z 0) = 0 then
|P (z 0)| = |λ||Q (z 0)| > |Q (z 0)|.Therefore |f (z 0)| = |P (z 0)/Q (z 0)| > 1 and so |z 0| < 1 by (∗). Now if Q (z 0) = 0, then |z 0| ≤ 1. So all zeros of g lie in |z | ≤ 1. So by Lucas’Theorem, g has all its zeros in |z | ≤ 1. So for no |λ| > 1 is
g (z ) = P (z ) − λQ (z ) = 0 where |z | > 1;
or in other words, P (z )/Q (z ) = λ where |λ| > 1 has no solution in|z | > 1. Hence |P (z )| ≤ |Q (z )| for |z | > 1. By taking limits, we have
|P
(z )| ≤ |Q
(z )| for |z | ≥ 1, and the result follows.Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 12 / 35
Bernstein’s Results
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Bernstein and de Bruijn
Bernstein’s Lemma was proved (as stated) by Sergei Bernstein [Lecons sur
les propietes extremales (Collection Borel) Paris, 1926]. A generalizationwas proven by Nicolass de Bruijn [Inequalities concerning polynomials inthe complex domain, Nederl. Akad. Wetensch. Proc. [Proceedings of the Royal Dutch Academy of Sciences ] 50 (1947), 1265–1272; Indagationes Mathematicae, Series A 9 (1947), 591–598] where |z | < 1 is replaced with
a convex domain D (a domain is an open connected set) and |z | = 1 isreplaced with the boundary of D , ∂ D .
Sergei N. Bernstein (1880–1968) Nicolaas de Bruijn (1918–2012)Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 13 / 35
Bernstein’s Results
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Bernstein’s Theorem.
Theorem
Let P be a polynomial of degree n. Then
max|z |=1
|P (z )| ≤ n max|z |=1
|P (z )|.
Proof. Let M = max|z |=1 |P (z )| and define Q (z ) = Mz
n
.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 14 / 35
Bernstein’s Results
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Bernstein’s Theorem.
Theorem
Let P be a polynomial of degree n. Then
max|z |=1
|P (z )| ≤ n max|z |=1
|P (z )|.
Proof.Let M = max|z |=1 |P (z )| and define Q (z ) = Mz
n
. Then (i)deg(P ) = deg(Q ), (ii) |P (z )| ≤ |Q (z )| = M for |z | = 1, and (iii) all zerosof Q lie in |z | ≤ 1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 14 / 35
Bernstein’s Results
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Bernstein’s Theorem.
Theorem
Let P be a polynomial of degree n. Then
max|z |=1
|P (z )| ≤ n max|z |=1
|P (z )|.
Proof.Let M = max|z |=1 |P (z )| and define Q (z ) = Mz
n
. Then (i)deg(P ) = deg(Q ), (ii) |P (z )| ≤ |Q (z )| = M for |z | = 1, and (iii) all zerosof Q lie in |z | ≤ 1. So by Bernstein’s Lemma
|P (z )| ≤ |Q (z )| for |z | = 1.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 14 / 35
Bernstein’s Results
B i ’ Th
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Bernstein’s Theorem.
Theorem
Let P be a polynomial of degree n. Then
max|z |=1
|P (z )| ≤ n max|z |=1
|P (z )|.
Proof.Let M = max|z |=1 |P (z )| and define Q (z ) = Mz
n
. Then (i)deg(P ) = deg(Q ), (ii) |P (z )| ≤ |Q (z )| = M for |z | = 1, and (iii) all zerosof Q lie in |z | ≤ 1. So by Bernstein’s Lemma
|P (z )| ≤ |Q (z )| for |z | = 1.
This implies that
max|z |=1
|P (z )| ≤ max|z |=1
|Q (z )| = max|z |=1
|nMz n−1| = nM = n max|z |=1
|P (z )|.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 14 / 35
Bernstein’s Results
B i ’ Th
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Bernstein’s Theorem.
Theorem
Let P be a polynomial of degree n. Then
max|z |=1
|P (z )| ≤ n max|z |=1
|P (z )|.
Proof. Let M = max|z |=1 |
P (z )|
and define Q (z ) = Mz n. Then (i)deg(P ) = deg(Q ), (ii) |P (z )| ≤ |Q (z )| = M for |z | = 1, and (iii) all zerosof Q lie in |z | ≤ 1. So by Bernstein’s Lemma
|P (z )| ≤ |Q (z )| for |z | = 1.
This implies that
max|z |=1
|P (z )| ≤ max|z |=1
|Q (z )| = max|z |=1
|nMz n−1| = nM = n max|z |=1
|P (z )|.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 14 / 35
Bernstein’s Results
M N
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Max Norms
For reasons to be made apparent later, we introduce the followingnotation:
P ∞ = max|z |=1
|P (z )|.
Bernstein’s Theorem can then be stated as:
Theorem
Let P be a polynomial of degree n. Then
P ∞ ≤ nP ∞.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 15 / 35
Bernstein’s Results
M N
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Max Norms
For reasons to be made apparent later, we introduce the followingnotation:
P ∞ = max|z |=1
|P (z )|.
Bernstein’s Theorem can then be stated as:
Theorem
Let P be a polynomial of degree n. Then
P ∞ ≤ nP ∞.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 15 / 35
Refinements of Bernstein
E d˝s L Th
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Erdos-Lax Theorem.
Theorem
Let P be a polynomial of degree n with P (z ) = 0 in |z | < 1. Then
P ∞ ≤ n
2P ∞.
Note. This result was conjectured by Erdos and proved by Lax. It is sharp
for P (z ) = αz n + β where |α| = |β |.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 16 / 35
Refinements of Bernstein
Erdos Lax Theorem
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Erdos-Lax Theorem.
Theorem
Let P be a polynomial of degree n with P (z ) = 0 in |z | < 1. Then
P ∞ ≤ n
2P ∞.
Note. This result was conjectured by Erdos and proved by Lax. It is sharp
for P (z ) = αz n + β where |α| = |β |.
Paul Erdos (1913–1996) Peter D. Lax (1926– )Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 16 / 35
Refinements of Bernstein
Erdos Lax Theorem
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Erdos-Lax Theorem.
Theorem
Let P be a polynomial of degree n with P (z ) = 0 in |z | < 1. Then
P ∞ ≤ n
2P ∞.
Note. This result was conjectured by Erdos and proved by Lax. It is sharp
for P (z ) = αz n + β where |α| = |β |.
Paul Erdos (1913–1996) Peter D. Lax (1926– )Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 16 / 35
Refinements of Bernstein
Malik’s Theorem
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Malik s Theorem.
TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < K, where K ≥ 1. Then
P ∞ ≤ n
1 + K P ∞.
Reference. Mohammad Malik, On the Derivative of a Polynomial,Journal of the London Mathematical Society , 1(2), 57–60 (1969).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 17 / 35
Refinements of Bernstein
Malik’s Theorem
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Malik s Theorem.
TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < K, where K ≥ 1. Then
P ∞ ≤ n
1 + K P ∞.
Reference. Mohammad Malik, On the Derivative of a Polynomial,Journal of the London Mathematical Society , 1(2), 57–60 (1969).
Note. This result is sharp for P (z ) =
z + K 1 + K
n
.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 17 / 35
Refinements of Bernstein
Malik’s Theorem
7/27/2019 Bernstein Beamer
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Malik s Theorem.
TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < K, where K ≥ 1. Then
P ∞ ≤ n
1 + K P ∞.
Reference. Mohammad Malik, On the Derivative of a Polynomial,Journal of the London Mathematical Society , 1(2), 57–60 (1969).
Note. This result is sharp for P (z ) =
z + K 1 + K
n
.
Note. With K = 1, Malik’s Theorem reduces to the Erdos-Lax Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 17 / 35
Refinements of Bernstein
Malik’s Theorem
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Malik s Theorem.
TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < K, where K ≥ 1. Then
P ∞ ≤ n
1 + K P ∞.
Reference. Mohammad Malik, On the Derivative of a Polynomial,Journal of the London Mathematical Society , 1(2), 57–60 (1969).
Note. This result is sharp for P (z ) =
z + K 1 + K
n
.
Note. With K = 1, Malik’s Theorem reduces to the Erdos-Lax Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 17 / 35
Refinements of Bernstein
Govil-Labelle Theorem
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Govil Labelle Theorem
TheoremIf P (z ) = an
nv =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then
P ∞ ≤ n
2 1 − 1
1 + 2
nn
v =1
1
K v −1 P ∞.
Reference. Narendra K. Govil and Gilbert Labelle, On Bernstein’sInequality, Journal of Mathematical Analysis and Applications , 126(2),
494–500 (1987).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 18 / 35
Refinements of Bernstein
Govil-Labelle Theorem
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Govil Labelle Theorem
TheoremIf P (z ) = an
nv =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then
P ∞ ≤ n
2 1 − 1
1 + 2
nn
v =1
1
K v −1 P ∞.
Reference. Narendra K. Govil and Gilbert Labelle, On Bernstein’sInequality, Journal of Mathematical Analysis and Applications , 126(2),
494–500 (1987).
Note. This result reduces to Malik’s Theorem if K v ≥ K ≥ 1 for all v ,and reduces to the Erdos-Lax Theorem if K v = 1 for some v .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 18 / 35
Refinements of Bernstein
Govil-Labelle Theorem
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Govil Labelle Theorem
TheoremIf P (z ) = an
nv =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then
P ∞ ≤ n
2 1 − 1
1 + 2
nn
v =1
1
K v −1 P ∞.
Reference. Narendra K. Govil and Gilbert Labelle, On Bernstein’sInequality, Journal of Mathematical Analysis and Applications , 126(2),
494–500 (1987).
Note. This result reduces to Malik’s Theorem if K v ≥ K ≥ 1 for all v ,and reduces to the Erdos-Lax Theorem if K v = 1 for some v .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 18 / 35
Refinements of Bernstein
Professor Qazi Ibadur Rahman
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Q b
Govil, Malik, and Labelle are each Ph.D. students of Professor Qazi I.Rahman of the University of Montreal: Mohammad Malik (Ph.D. 1967),Narendra Govil (Ph.D. 1968), and Gilbert Labelle (Ph.D. 1969).
Narendra Govil, Qazi Rahman, and Robert Gardner at the 2005 FallSoutheast Section AMS Meeting, October 2005, at ETSU.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 19 / 35
L
p
Spaces
Normed Linear Spaces
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p
DefinitionLet X be a set of [equivalence classes of] functions. Then X is a linear space if for all f , g ∈ X [or for all equivalence classes [f ], [g ] ∈ X ] andα, β ∈ R, αf + β g ∈ X [or α[f ] + β [g ] ∈ X ].
Definition
Let X be a linear space. A real-valued functional (i.e., a function with X as its domain and R as its codomain) · on X is a norm if for allf , g ∈ X and for all α ∈ R:
(1) f + g ≤ f + g (Triangle Inequality).(2) αf = |α|f (Positive Homogeneity).
(3) f ≥ 0 and f = 0 if and only if f = 0.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 20 / 35
L
p
Spaces
Real Lp Spaces
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p
Definition
A normed linear space is a linear space X with a norm · .
Definition
Let E be a measurable set of real numbers and let 1 ≤ p <∞. Define
Lp
(E ) to be the set of [equivalence classes of] functions for which
E
|f |p <∞.
Definition
For measurable set E , Lp (E ) is a normed linear space with norm
f p =
E
|f |p <
1/p
. In fact, Lp (E ) is a complete normed linear space
(i.e., a Banach space ).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 21 / 35
Lp
Spaces
Lp Norm
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DefinitionLet P be a complex polynomial. For p ≥ 1 the Lp norm of P is
P
p = 1
2π
2π
0 |P (e i θ)
|p d θ
1/p
.
Theorem
If we let p →∞, then we find that
limp →∞
P p = limp →∞
1
2π
2π
0|P (e i θ)|p d θ
1/p
= max|z |=1
|P (z )| ≡ P ∞.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 22 / 35
Bernstein inLp
Space
Zygmund-Arestov Theorem
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Theorem
Let P be a polynomial of degree n. Then for 1 ≤ p ≤ ∞, P p ≤ nP p .
References. A. Zygmund, A Remark on Conjugate Series, Proceedings of the London Mathematical Society (2) 34, 392–400 (1932). V. V. Arestov,On Inequalities for Trigonometric Polynomials and Their Derivatives, Izv.Akad. Nauk SSSR Ser. Mat. 45, 3–22 (1981) [in Russian]; Math.USSR-Izv. 18, 1–17 (1982) [in English].
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 23 / 35
Bernstein in Lp Space
DeBruijn’s Theorem.
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TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < 1. Then for 1 ≤ p ≤ ∞,
P p ≤ n
1 + z
p P p .
Reference. Nicolass de Bruijn, Inequalities concerning polynomials in thecomplex domain, Nederl. Akad. Wetensch. Proc. 50 (1947), 1265–1272;Indag. Math. 9 (1947), 591–598.
Note. If we let p →∞, DeBruijn’s Theorem reduces to the Erdos-LaxTheorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 24 / 35
Bernstein in Lp Space
DeBruijn’s Theorem.
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TheoremLet P be a polynomial of degree n with P (z ) = 0 in |z | < 1. Then for 1 ≤ p ≤ ∞,
P p ≤ n
1 + z
p P p .
Reference. Nicolass de Bruijn, Inequalities concerning polynomials in thecomplex domain, Nederl. Akad. Wetensch. Proc. 50 (1947), 1265–1272;Indag. Math. 9 (1947), 591–598.
Note. If we let p →∞, DeBruijn’s Theorem reduces to the Erdos-LaxTheorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 24 / 35
Bernstein in Lp Space
Gardner-Govil Theorem
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Theorem
If P (z ) = an
nv =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for p ≥ 1,
P p ≤ n
t 0 + z p P p
where t 0 =Pn
v =1
K v
K v −
1Pnv =1
1K v −1
= 1 + nPn
v =11
K v −1.
Reference. Robert Gardner and Narendra Govil, Inequalities Concerning
the Lp
Norm of a Polynomial and its Derivative, Journal of Mathematical Analysis and Applications , 179(1) (1993), 208–213.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 25 / 35
Bernstein in Lp Space
Gardner-Govil Theorem
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Theorem
If P (z ) = an
nv =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for p ≥ 1,
P p ≤ n
t 0 + z p P p
where t 0 =Pn
v =1
K v
K v −
1Pnv =1
1K v −1
= 1 + nPn
v =11
K v −1.
Reference. Robert Gardner and Narendra Govil, Inequalities Concerning
the Lp
Norm of a Polynomial and its Derivative, Journal of Mathematical Analysis and Applications , 179(1) (1993), 208–213.
Note. If K v = 1 for any v , this result reduces to Debruijn’s Theorem. If p
→∞, then this result reduces to the Govil-Labelle Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 25 / 35
Bernstein in Lp Space
Gardner-Govil Theorem
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Theorem
If P (z ) = ann
v =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for p ≥ 1,
P p ≤ n
t 0 + z p P p
where t 0 =Pn
v =1
K v
K v −
1Pnv =1
1K v −1
= 1 + nPn
v =11
K v −1.
Reference. Robert Gardner and Narendra Govil, Inequalities Concerning
the Lp
Norm of a Polynomial and its Derivative, Journal of Mathematical Analysis and Applications , 179(1) (1993), 208–213.
Note. If K v = 1 for any v , this result reduces to Debruijn’s Theorem. If p
→∞, then this result reduces to the Govil-Labelle Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 25 / 35
Other Types of Bernstein Inequalities
Lp “Quantities”, 0 ≤ p < 1
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Definition
Let P be a polynomial. For 0 < p < 1 define the “Lp quantity ” of P as
P p =
1
2π
2π
0|P (e i θ)|p d θ
1/p
.
Note. In the event that we let p → 0+, we find that
limp →0+
P p = exp
1
2π
2π
0log |P (e i θ)| d θ
.
We define this as the “L0 quantity ” of P , denoted P 0.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 26 / 35
Other Types of Bernstein Inequalities
Lp “Quantities”, 0 ≤ p < 1
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Definition
Let P be a polynomial. For 0 < p < 1 define the “Lp quantity ” of P as
P p =
1
2π
2π
0|P (e i θ)|p d θ
1/p
.
Note. In the event that we let p → 0+, we find that
limp →0+
P p = exp
1
2π
2π
0log |P (e i θ)| d θ
.
We define this as the “L0 quantity ” of P , denoted P 0.Note. For 0 ≤ p < 1, the Lp quantity of a polynomial does not satisfy theTriangle Inequality. In fact (see Halsey Royden’s Real Analysis , 3rdEdition, page 120): f + g p ≥ f p + g p .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 26 / 35
Other Types of Bernstein Inequalities
Lp “Quantities”, 0 ≤ p < 1
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Definition
Let P be a polynomial. For 0 < p < 1 define the “Lp quantity ” of P as
P p =
1
2π
2π
0|P (e i θ)|p d θ
1/p
.
Note. In the event that we let p → 0+, we find that
limp →0+
P p = exp
1
2π
2π
0log |P (e i θ)| d θ
.
We define this as the “L0 quantity ” of P , denoted P 0.Note. For 0 ≤ p < 1, the Lp quantity of a polynomial does not satisfy theTriangle Inequality. In fact (see Halsey Royden’s Real Analysis , 3rdEdition, page 120): f + g p ≥ f p + g p .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 26 / 35
Other Types of Bernstein Inequalities
Theorem
If P(z) an (z z ) (a 0) and |z | ≥ K ≥ 1 Then for
7/27/2019 Bernstein Beamer
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If P (z ) = an
v =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for
0 ≤ p < 1,
P
p ≤n
t 0 + z p P p
where t 0 =
Pnv =1
K v K v −1Pn
v =11
K v −1
= 1 + nPn
v =11
K v −1
.
Reference. Robert Gardner and Narendra Govil, An Lp Inequality for aPolynomial and its Derivative, Journal of Mathematical Analysis and Applications , 193(2), 490–496 (1995); 194(3), 720–726 (1995).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 27 / 35
Other Types of Bernstein Inequalities
Theorem
If P(z) = an (z z ) (a = 0) and |z | ≥ K ≥ 1 Then for
7/27/2019 Bernstein Beamer
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If P (z ) = an
v =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for
0 ≤ p < 1,
P
p ≤n
t 0 + z p P p
where t 0 =
Pnv =1
K v K v −1Pn
v =11
K v −1
= 1 + nPn
v =11
K v −1
.
Reference. Robert Gardner and Narendra Govil, An Lp Inequality for aPolynomial and its Derivative, Journal of Mathematical Analysis and Applications , 193(2), 490–496 (1995); 194(3), 720–726 (1995).
Note. With some K v = 1, we can extend DeBruijn to 0 ≤ p < 1: Let P bea polynomial of degree n with P (z ) = 0 in |z | < 1. Then for 0 ≤ p ≤ ∞,
P p ≤ n
1 + z p P p .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 27 / 35
Other Types of Bernstein Inequalities
Theorem
If P(z) = an (z z ) (a = 0) and |z | ≥ K ≥ 1 Then for
7/27/2019 Bernstein Beamer
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If P (z ) = an
v =1(z − z v ) (an = 0), and |z v | ≥ K v ≥ 1. Then for
0 ≤ p < 1,
P
p ≤n
t 0 + z p P p
where t 0 =
Pnv =1
K v K v −1Pn
v =11
K v −1
= 1 + nPn
v =11
K v −1
.
Reference. Robert Gardner and Narendra Govil, An Lp Inequality for aPolynomial and its Derivative, Journal of Mathematical Analysis and Applications , 193(2), 490–496 (1995); 194(3), 720–726 (1995).
Note. With some K v = 1, we can extend DeBruijn to 0 ≤ p < 1: Let P bea polynomial of degree n with P (z ) = 0 in |z | < 1. Then for 0 ≤ p ≤ ∞,
P p ≤ n
1 + z p P p .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 27 / 35
Other Types of Bernstein Inequalities
A New Linear Space of Polynomials
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Definition
Let P n,m denote the set of all polynomials of the form
P (z ) = a0 +
nv =m
av z v
= a0 + amz m
+ am+1z m+1
+ · · · + anz n
.
Note. P n,m is a linear space and P n,1 = P n.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 28 / 35
Other Types of Bernstein Inequalities
A New Linear Space of Polynomials
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Definition
Let P n,m denote the set of all polynomials of the form
P (z ) = a0 +
nv =m
av z v
= a0 + amz m
+ am+1z m+1
+ · · · + anz n
.
Note. P n,m is a linear space and P n,1 = P n.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 28 / 35
Other Types of Bernstein Inequalities
Chan-Malik Theorem
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Theorem
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + K mP ∞.
Reference. T. Chan and M. Malik, On Erdos-Lax Theorem, Proceedings of the Indian Academy of Sciences 92, 191-193 (1983).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 29 / 35
Other Types of Bernstein Inequalities
Chan-Malik Theorem
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Theorem
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + K mP ∞.
Reference. T. Chan and M. Malik, On Erdos-Lax Theorem, Proceedings of the Indian Academy of Sciences 92, 191-193 (1983).
Note. With m = 1, the Chan-Malik Theorem reduces to Malik’s Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 29 / 35
Other Types of Bernstein Inequalities
Chan-Malik Theorem
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Theorem
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + K mP ∞.
Reference. T. Chan and M. Malik, On Erdos-Lax Theorem, Proceedings of the Indian Academy of Sciences 92, 191-193 (1983).
Note. With m = 1, the Chan-Malik Theorem reduces to Malik’s Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 29 / 35
Other Types of Bernstein Inequalities
Qazi’s Theorem
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Theorem
If P (z ) = a0 +
nv =m av z v ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + s 0P ∞,
where s 0 = K m+1m
|am
|K m−1 + n
|a0
|n|a0| + m|am|K m+1.
Reference. M. Qazi, On the Maximum Modulus of Polynomials,
Proceedings of the American Mathematical Society 115
, 337–343 (1992).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 30 / 35
Other Types of Bernstein Inequalities
Qazi’s Theorem
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Theorem
If P (z ) = a0 +
nv =m av z v ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + s 0P ∞,
where s 0 = K m+1m
|am
|K m−1 + n
|a0
|n|a0| + m|am|K m+1.
Reference. M. Qazi, On the Maximum Modulus of Polynomials,Proceedings of the American Mathematical Society 115, 337–343 (1992).
Note. Qazi worked independently of Chan and Malik. Sincem|am|K m ≤ n|a0| (as Qazi explains in his paper), this result implies theChan-Malik Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 30 / 35
Other Types of Bernstein Inequalities
Qazi’s Theorem
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Theorem
If P (z ) = a0 +
nv =m av z v ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then
P ∞ ≤ n
1 + s 0P ∞,
where s 0 = K m+1m
|am
|K m−1 + n
|a0
|n|a0| + m|am|K m+1.
Reference. M. Qazi, On the Maximum Modulus of Polynomials,Proceedings of the American Mathematical Society 115, 337–343 (1992).
Note. Qazi worked independently of Chan and Malik. Sincem|am|K m ≤ n|a0| (as Qazi explains in his paper), this result implies theChan-Malik Theorem.
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 30 / 35
Other Types of Bernstein Inequalities
Gardner-Weems Theorem
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Theorem
If P (z ) = a0 +n
v =m av z v ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then for 0 ≤ p ≤ ∞
P p ≤ n
s 0 + z
p P p ,
where s 0 = K m+1
m|am|K m−1 + n|a0|n|a0| + m|am|K m+1
.
Reference. Robert Gardner and Amy Weems, A Bernstein Type Lp
Inequality for a Certain Class of Polynomials, Journal of Mathematical Analysis and Applications , 219, 472–478 (1998).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 31 / 35
Other Types of Bernstein Inequalities
Gardner-Weems Theorem
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Theorem
If P (z ) = a0 +n
v =m av z v ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then for 0 ≤ p ≤ ∞
P p ≤ n
s 0 + z
p P p ,
where s 0 = K m+1
m|am|K m−1 + n|a0|n|a0| + m|am|K m+1
.
Reference. Robert Gardner and Amy Weems, A Bernstein Type Lp
Inequality for a Certain Class of Polynomials, Journal of Mathematical Analysis and Applications , 219, 472–478 (1998).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 31 / 35
Other Types of Bernstein Inequalities
Gardner-Weems Corollary
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Note. With p = ∞, this reduces to Qazi’s Theorem, and further reduces
to the Chan-Malik Theorem. If p = ∞ and m = 1 this reduces to Malik’sTheorem, and if in addition K = 1 then it reduces to the Erdos-LaxTheorem.
Corollary
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then for 1 ≤ p ≤ ∞P p ≤ n
K m + z p P p .
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 32 / 35
Other Types of Bernstein Inequalities
Gardner-Weems Corollary
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Note. With p = ∞, this reduces to Qazi’s Theorem, and further reduces
to the Chan-Malik Theorem. If p = ∞ and m = 1 this reduces to Malik’sTheorem, and if in addition K = 1 then it reduces to the Erdos-LaxTheorem.
Corollary
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then for 1 ≤ p ≤ ∞P p ≤ n
K m + z p P p .
Note. This corollary follows since m|am|K m ≤ n|a0| (as Qazi explains).Of course, the corollary also holds for 0 ≤ p < 1 (though does not theninvolve norms).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 32 / 35
Other Types of Bernstein Inequalities
Gardner-Weems Corollary
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Note. With p = ∞, this reduces to Qazi’s Theorem, and further reduces
to the Chan-Malik Theorem. If p = ∞ and m = 1 this reduces to Malik’sTheorem, and if in addition K = 1 then it reduces to the Erdos-LaxTheorem.
Corollary
If P ∈ P n,m and P (z ) = 0 for |z | < K where K ≥ 1, then for 1 ≤ p ≤ ∞P p ≤ n
K m + z p P p .
Note. This corollary follows since m|am|K m ≤ n|a0| (as Qazi explains).Of course, the corollary also holds for 0 ≤ p < 1 (though does not theninvolve norms).
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 32 / 35
Other Types of Bernstein Inequalities
Amy Weems
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Reference. Robert Gardner and Amy Weems, A Bernstein Type Lp
Inequality for a Certain Class of Polynomials, Journal of Mathematical Analysis and Applications , 219, 472–478 (1998).
Amy Vaughan Weems(9/21/1966 – 4/7/1998)
Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 33 / 35
Other Types of Bernstein Inequalities
Amy Weems
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Reference. Robert Gardner and Amy Weems, A Bernstein Type Lp
Inequality for a Certain Class of Polynomials, Journal of Mathematical Analysis and Applications , 219, 472–478 (1998).
Amy Vaughan Weems(9/21/1966 – 4/7/1998)
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Conclusion
Upcoming Classes
ETSU ill ff I t d ti t F ti l A l i (MATH 5740)
7/27/2019 Bernstein Beamer
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ETSU will offer Introduction to Functional Analysis (MATH 5740)during Summer 2013 Term 1. Linear spaces and Lp norms will be a centraltopic. The prerequisite is Analysis 1 (MATH 4217/5217). Duringacademic 2013–14, Complex Analysis 1 and 2 (MATH 5510/5520) willbe offered. Analytic functions and complex integration will be covered inpart 1. The prerequisite is also Analysis 1 and no previous experience with
complex variables is assumed. Complex Analysis 2 will cover the MaximumModulus Theorem and analytic continuation.
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References
References
Ph t h f th ti i i il f htt // hi t t d k/
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Photographs of mathematicians are primarily from http://www-history.mcs.st-and.ac.uk/ .
1 Sergei Bernstein, Lecons sur les propietes extremales (Collection Borel) Paris (1926).
2 Ralph Boas, Inequalities for the Derivatives of Polynomials, Mathematics Magazine 42(4), 165–174 (1969).3 Nicolass de Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch.
Proc.[Proceedings of the Royal Dutch Academy of Sciences ] 50 (1947), 1265–1272; Indagationes Mathematicae, Series A 9 (1947), 591–598.
4 T. Chan and M. Malik, On Erdos-Lax Theorem, Proceedings of the Indian Academy of Sciences 92, 191-193 (1983).
5 Robert Gardner and Narendra Govil, Inequalities Concerning the Lp Norm of a Polynomial and its Derivative, Journal of Mathematical Analysis and Applications , 179(1) (1993), 208–213.
6Robert Gardner and Narendra Govil, An
Lp
Inequality for a Polynomial and its Derivative,Journal of Mathematical
Analysis and Applications , 193(2), 490–496 (1995); 194(3), 720–726 (1995).
7 Robert Gardner and Amy Weems, A Bernstein Type Lp Inequality for a Certain Class of Polynomials, Journal of Mathematical Analysis and Applications , 219, 472–478 (1998).
8 Narendra K. Govil and Gilbert Labelle, On Bernstein’s Inequality, Journal of Mathematical Analysis and Applications ,126(2), 494–500 (1987).
9 Mohammad Malik, On the Derivative of a Polynomial, Journal of the London Mathematical Society , 1(2), 57–60 (1969).
10 Morris Marden, Geometry of Polynomials , American Mathematical Society Math Surveys, 3 (1985).
11 M. Qazi, On the Maximum Modulus of Polynomials, Proceedings of the American Mathematical Society 115, 337–343(1992). Sharpness and Erdos-Lax
12 A. Zygmund, A Remark on Conjugate Series, Proceedings of the London Mathematical Society (2) 34, 392–400 (1932).V. V. Arestov, On Inequalities for Trigonometric Polynomials and Their Derivatives, Izv. Akad. Nauk SSSR Ser. Mat.45, 3–22 (1981) [in Russian]; Math. USSR-Izv. 18, 1–17 (1982) [in English].
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