Cases of Equality for a Class of Bound-Preserving Operators over Pn

6
Computational Methods and Function Theory Volume 4 (2004), No. 1, 183–188 Cases of Equality for a Class of Bound-Preserving Operators over P n Richard Fournier (Communicated by Stephan Ruscheweyh) Dedicated to the memory of Walter Hengartner Abstract. Let D denote the unit disc of the complex plane and P n the class of polynomials of degree at most n with complex coefficients. We give a new proof of p(z) zp (z) n + zp (z) n ≤|p| D , z D,p ∈P n , together with a complete discussion of all cases of equality. We also discuss an extension, due to Ruscheweyh, of the above inequality. Keywords. Complex polynomials, Bernstein inequality, generalizations of Bernstein inequality. 2000 MSC. Primary: 41A17. 1. Introduction Let D denote the unit disc {z C : |z | < 1} of the complex plane C and H(D) the set of functions analytic on D. We define for f ∈H(D) |f | D := sup zD |f (z )|. Let also P n denote the set of polynomials p(z ) := n k=0 a k (p)z k of degree at most n with complex coefficients. The inequality (valid for any p ∈P n ) (1) |zp (z ) np(z )| + |zp (z )|≤ n|p| D , |z |≤ 1, is a well-known refinement of the classical Bernstein inequality (2) |p | D n|p| D . Although it is not clear at all when (1) was first obtained, many proofs of it and its generalizations are now known (see [3, p. 28], [4, p. 524], [5, p. 128] or [6, Received May 27, 2004, in revised form July 12, 2004. ISSN 1617-9447/$ 2.50 c 2004 Heldermann Verlag

Transcript of Cases of Equality for a Class of Bound-Preserving Operators over Pn

Page 1: Cases of Equality for a Class of Bound-Preserving Operators over Pn

Computational Methods and Function TheoryVolume 4 (2004), No. 1, 183–188

Cases of Equality for a Classof Bound-Preserving Operators over Pn

Richard Fournier

(Communicated by Stephan Ruscheweyh)

Dedicated to the memory of Walter Hengartner

Abstract. Let D denote the unit disc of the complex plane and Pn the classof polynomials of degree at most n with complex coefficients. We give a newproof of ∣∣∣∣p(z)− zp′(z)

n

∣∣∣∣ +∣∣∣∣zp′(z)

n

∣∣∣∣ ≤ |p|D, z ∈ D, p ∈ Pn,

together with a complete discussion of all cases of equality. We also discussan extension, due to Ruscheweyh, of the above inequality.

Keywords. Complex polynomials, Bernstein inequality, generalizations ofBernstein inequality.

2000 MSC. Primary: 41A17.

1. Introduction

Let D denote the unit disc {z ∈ C : |z| < 1} of the complex plane C and H(D)the set of functions analytic on D. We define for f ∈ H(D)

|f |D := supz∈D

|f(z)|.

Let also Pn denote the set of polynomials p(z) :=∑n

k=0 ak(p)zk of degree at mostn with complex coefficients. The inequality (valid for any p ∈ Pn)

(1) |zp′(z)− np(z)|+ |zp′(z)| ≤ n|p|D, |z| ≤ 1,

is a well-known refinement of the classical Bernstein inequality

(2) |p′|D ≤ n|p|D.

Although it is not clear at all when (1) was first obtained, many proofs of it andits generalizations are now known (see [3, p. 28], [4, p. 524], [5, p. 128] or [6,

Received May 27, 2004, in revised form July 12, 2004.

ISSN 1617-9447/$ 2.50 c© 2004 Heldermann Verlag

Page 2: Cases of Equality for a Class of Bound-Preserving Operators over Pn

184 R. Fournier CMFT

pp. 151–152]). It does not seem to be known, however, for which polynomialsequality holds in (1). Sheil-Small [6] has remarked that

|up′(u)− np(u)|+ |up′(u)| = n|p|Dif p ∈ Pn, |u| = 1 and |p(u)| = |p|D. It is also rather obvious that

|uq′(u)− nq(u)|+ |uq′(u)| = |q|D, |u| = 1,

for polynomials q(z) ≡ Azn + B, A, B ∈ C. Our first result in this papercontains a new proof of (1). We shall also obtain all possible cases of equalityin (1) and (2).

For functions f(z) :=∑∞

n=0 an(f)zn and g(z) :=∑∞

n=0 an(g)zn in H(D), theHadamard product

f ∗ g(z) :=∞∑

n=0

an(f)an(g)zn

also belongs to H(D). A striking extension of (1) has been obtained by Rusche-weyh [5, p. 128]. In what follows Qn shall always designate a subclass of Pn−1

such that

q ∈ Qn ⇔ q(0) = 1 and inf|u|≤1

Re q(u) ≥ 1

2.

Theorem A. Let q ∈ Qn and q(z) := znq(1/z). Then

(3) |q ∗ p(z)|+ |q ∗ p(z)| ≤ |p|D, p ∈ Pn, |z| ≤ 1.

This result contains a lot of information. The case q(z) ≡ ∑nk=0(1−k/n)zk leads

to (1) while the case q(z) ≡ 1 leads to the inequality of Visser (see [2]). Ourmethod can be used to decide some cases of equality in (3).

Theorem 1. Let q ∈ Qn with inf |u|≤1 Re q(u) > 1/2. Then the equality

|q ∗ p(u)|+ |q ∗ p(u)| = |p|Dholds for some u ∈ D and p ∈ Pn if and only if |u| = 1 and p(z) ≡ Azn + B,A, B ∈ C.

Theorem 2. Equality holds in (1) if and only if p ∈ Pn, |z| = 1 and |p(z)| = |p|Dor else p(u) ≡ Aun + B, A, B ∈ C.

2. Proof of Theorem 1

Let q(z) =∑n

k=0 ckzk ∈ Qn, c0 = 1, cn = 0. We shall prove that, for any real θ,

(4) q(z) + eiθq(z) =n−1∑j=0

λj

1− eiθ/nwjz+ O(zn), wj := e2ijπ/n,

Page 3: Cases of Equality for a Class of Bound-Preserving Operators over Pn

4 (2004), No. 1 Cases of Equality for a Class of Bound-Preserving Operators over Pn 185

where the numbers λj = λj(q, θ) are non-negative with∑n−1

j=0 λj = 1. In fact (4)leads to the linear system

(5)n−1∑j=0

wjkλj =

(ck + eiθcn−k

)e−ikθ/n, k = 0, . . . , n− 1,

i.e.,

(6) V T Λ = C

where V T is the transpose of the Vandermonde matrix V formed with the set ofnodes {wj}n−1

j=0 , Λ := (λ1, λ2, . . . , λn)T and C is the column vector formed withthe constants on the right of (5). Let W (z) := zn − 1. Then

W (z)

(z − wk)W ′(wk)=

n−1∑t=0

wkt

nzt

and the (k+1)th column of V −1 is (wk0/n, wk

1/n, . . . , wkn−1/n)T , 0 ≤ k ≤ n−1,

([1, p. 13]). Therefore

(7) λj =1

n

n−1∑t=0

(ct + eiθcn−t

) (wj e−iθ/n

)t=

2

n

(Re q

(wj e−iθ/n

)− 1

2

).

Let us now assume that for some p ∈ Pn, θ ∈ R and z ∈ ∂D

|q ∗ p(z) + eiθq ∗ p(z)| = |q ∗ p(z)|+ |q ∗ p(z)| = |p|D.

Then by (4),

(8) |p|D =

∣∣∣∣∣n−1∑j=0

λjp(eiθ/nwjz

)∣∣∣∣∣ ≤n−1∑j=0

λj

∣∣p(eiθ/nwjz)∣∣ ≤ (

n−1∑j=0

λj

)|p|D = |p|D.

According to our hypothesis we have λj > 0, j = 0, . . . , n− 1, and equality musthold everywhere in (8). In particular there must exist a real number ϕ such that

p(eiθ/nwjz

)= |p|Deiϕ, j = 0, . . . , n− 1.

It is readily seen that

p(z) ≡ eiϕ|p|D −K(zn − eiθ), K ∈ C,

i.e., p(u) ≡ Aun + B, A, B ∈ C. This completes the proof of Theorem 1.

3. Proof of Theorem 2

Let us now assume that for some p ∈ Pn and z ∈ D equality holds in (1). Wemay clearly claim that p is non-constant and by the Maximum Principle, |z| = 1.There also exists a real number θ such that

(9) |p|D =

∣∣∣∣p(z)− zp′(z)

n

∣∣∣∣ +

∣∣∣∣zp′(z)

n

∣∣∣∣ =

∣∣∣∣p(z)− (1− eiθ)zp′(z)

n

∣∣∣∣ .

Page 4: Cases of Equality for a Class of Bound-Preserving Operators over Pn

186 R. Fournier CMFT

We first assume that θ = 0 (mod 2π); this means that the polynomial p attainsits maximum modulus on ∂D at z, i.e. |p(z)| = |p|D. If θ �= 0, then by (4) and (7)with q(z) ≡ ∑n

k=0(1− k/n)zk,

λk =|1− eiθ|2

4n2 sin2(

θ+2kπ2n

) > 0, 0 ≤ k ≤ n− 1.

and ∣∣∣∣∣n−1∑k=0

λkp(eiθ/nwkz

)∣∣∣∣∣ =n−1∑k=0

λk

∣∣p(eiθ/nwkz)∣∣ = |p|D,

By the equality case of the triangle inequality, there must exist a real number ϕsuch that

p(eiθ/nwkz

)= eiϕ|p|D, 0 ≤ k ≤ n− 1,

i.e., p(u) ≡ Aun + B, A, B ∈ C. This completes the proof of Theorem 2.

We shall end this section by stating without proof a variant of (1) which is alsoa consequence of (4). The inequality

Re

(p(z)− zp′(z)

n

)+

∣∣∣∣zp′(z)

n

∣∣∣∣ ≤ max|u|≤1

Re p(u), |z| ≤ 1, p ∈ Pn,

follows, just like (1) does, from Laguerre’s Theorem [4, p. 98]. Our method yieldsthat equality holds there if and only if |z| = 1 and Re p(z) = max|u|≤1 Re p(u) orelse p(u) ≡ Aun + B, A, B ∈ C. Moreover it follows that

|p′|D ≤ n

2

(max|u|≤1

Re p(u)− min|u|≤1

Re p(u)

)≤ n|Re p|D, p ∈ Pn,

i.e., we obtain a refinement of Szego’s inequality

|p′|D ≤ n|Re p|D, p ∈ Pn.

Further, equality holds in Szego’s inequality if and only if p(z) ≡ Azn + B,A, B ∈ C, Re(B) = 0.

4. Conclusion

Theorem A has a rather wide scope. For example the choice q(z) = 1 + εzk with|ε| < 1/2 and 0 < k < n yields

(10) |a0(p) + εak(p)|+ |εan−k(p) + an(p)| ≤ |p|D, p ∈ Pn

and equality holds in (10) only for polynomials p of type p(u) ≡ Aun +B. This isan interesting extension of a now classical result due to Visser (see [2] for relevantreferences). It has also been pointed out in [5, p. 129] how useful Theorem 2 canbe in the study of certain extremal properties of self-reciprocal polynomials (i.e.,

polynomials p ∈ Pn which satisfy p(z) ≡ znp(1/z)).

Page 5: Cases of Equality for a Class of Bound-Preserving Operators over Pn

4 (2004), No. 1 Cases of Equality for a Class of Bound-Preserving Operators over Pn 187

Theorem 2 (which corresponds to q(z) ≡ ∑nk=0(1 − k/n)zk) is not, however, a

consequence of Theorem 1 because

infθ∈[0,2π]

Re q(eiθ) = infθ∈[0,2π]

1

2+

1

2n

sin2(nθ/2)

sin2(θ/2)=

1

2.

There are indeed “too many” cases of equality in Theorem 2 and one may ask ifthat will also be the case for any inequality as

|Q ∗ p(z)|+ |Q ∗ p(z)| ≤ |p|D, p ∈ Pn,

where infθ∈[0,2π] Re Q(eiθ) = 1/2? We show that this is not true by consideringQ(z) = 1 + z/2: we show that

(11)

∣∣∣∣a0(p) +1

2a1(p)

∣∣∣∣ +

∣∣∣∣12an−1(p) + an(p)

∣∣∣∣ ≡ |p|Dholds for some p ∈ Pn if and only if p satisfies the conclusion of Theorem 1. Theproof of Theorem 1 (more precisely: the fact that at most one element of {λj}n−1

j=0

is zero) shows that any polynomial p as in (11) must satisfy

(12) p(z) ≡ Meiϕ − zn − eiθ

z − wjeiθ/n(Az + B),

where 1 ≤ j ≤ n, ϕ, θ ∈ [0, 2π], A, B ∈ C and M = |p|D. We can prove that nosuch polynomial exists unless

p(z) ≡ (M − |ρ|)eiϕ + ρzn, ρ ∈ C, 0 ≤ |ρ| ≤M.

We only sketch the proof. Starting from (12) we may clearly assume that

p(z) ≡ 1− zn − 1

z − 1(Az + B), A, B ∈ C, |p|D = 1.

Two applications of a variant of Visser’s inequality [2] yield

p(z) ≡ 1−Bzn − 1

z − 1(ξz + 1), B ∈ C,

with

0 < |ξ| < 1 and |1 + ξ| ≤ 1.

By convexity we may restrict ourselves to the case where B ≥ 0 and −1 ≤ ξ < 0,i.e., the existence of non-constant polynomials as in (12) amounts to the factthat

(13) inf|z|≤1

Re1− z

(1− zn)(1 + ξz)> 0, −1 ≤ ξ < 0.

A study of the limits

limz→e2ijπ/n

1− z

(1− zn)(1 + ξz),

n

2< j < n,

finally shows that (13) cannot hold unless ξ = −1.

Page 6: Cases of Equality for a Class of Bound-Preserving Operators over Pn

188 R. Fournier CMFT

References

1. E. W. Cheney, Introduction to Approximation Theory, Chelsea Publ. Co., New-York, 1982.2. D. Dryanov and R. Fournier, Bound preserving operators over classes of polynomials, East

J. Approx. 8 (2002), 327–353.3. Q. I. Rahman and G. Schmeisser, Les inegalites de Markoff et de Bernstein, Les Presses

de l’Universite de Montreal, Montreal, 1983.4. , Analytic Theory of Polynomials, Oxford Univ. Press, Oxford, 2002.5. St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l’Universite

de Montreal, Montreal, 1982.6. T. Sheil-Small, Complex Polynomials, Cambridge Univ. Press, Cambridge, 2002.

Richard Fournier E-mail: [email protected]: Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128, succ.Centre-ville, Montreal, Qc H3C 3J7, Canada.