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

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,

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

∣∣∣∣ .

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)).

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.

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: fournier@crm.umontreal.caAddress: Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128, succ.Centre-ville, Montreal, Qc H3C 3J7, Canada.