Bernstein Beamer

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Bernstein Inequalities for Polynomials Robert “Dr. Bob” Gardner March 8, 2013 (Prepared in Beamer!) Robert “Dr. Bo b” Ga rdner ()  Bernstein Inequalities for Polynomials  March 8, 2013 1 / 35

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.

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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

|> 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

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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

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

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 

|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  = 1

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

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

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

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

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

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

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

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

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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  ≤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

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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  ≤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

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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  ≤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)

Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 33 / 35

Conclusion

Upcoming Classes

ETSU ill ff I t d ti t F ti l A l i (MATH 5740)

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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.

Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 34 / 35

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].

Robert “Dr. Bob” Gardner () Bernstein Inequalities for Polynomials March 8, 2013 35 / 35