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