BOUNDING ZEROS OF H^{2}

FUNCTIONS VIA CONCENTRATIONS

maria girardi

J. Math. Anal. Appl. 183 (1994) 605–612

Abstract. It is well-known that the zeros {z_{j}} of a function in the classical Hardy space H^{2} satisfy ∑ 1−|z_{j}| < ∞ ; however, this sum can be arbitrarily large. We shall bound this sum by a constant that depends on the concentration of the function, a concept introduced by Beauzamy and Enflo.

1. INTRODUCTION

Consider a function f : D → C in the classical Hardy space H^{2}(D) where D is the open unit disk in the complex plane C. It is well-known that the zeros {z_{j}} of f satisfy ∑ 1 − |z_{j}| < ∞. However, this sum can be arbitrarily large as seen by considering an appropriate Blaschke product.

Fix 1 ≤ p ≤ 2 and consider the subset A_{p} of H^{2} where

A_{p} = {f ∈ H^{p}(D) : f(z) = ∑_{j≥0} a_{j}z^{j} and ∑_{j≥0} |a_{j}|^{p} < ∞}.

Of course, A_{2} is just H^{2} and A_{1} is the usual algebra A_{+}(T). For a function in A_{p} with zeros {z_{j}}, we shall bound ∑ 1 − |z_{j}| by a constant that depends on the concentration of the function, a notation introduced by Beauzamy and Enflo [BE].

Now to recall basic facts and fix notation. For

f(z) = ∑_{j≥0} a_{j}z^{j} ∈ A_{p},

the l_{p}-norm of f is

∥ f ∥_{p} = [ ∑_{j≥0} |a_{j}|^{p} ]^{1/p},

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Supported in part by the C.N.R.S. (France) and the N.S.F. (U.S.A.), by contracts D.G.A./- D.R.E.T. no 89/1377, E.T.C.A./C.R.E.A./20367/91 (Ministry of Defense, France) and by research contract EERP-FR 22, Digital Equipment Corporation. Research conducted while at the Institut de Calcul Mathématique, Université de Paris VII, 2 place Jussieu, 75251 Paris Cedex 05, France.

and the k-th partial sum s_{k}(f) of f is

[s_{k}(f)](z) = ∑_{j=0}^{k} a_{j}z^{j}.

The function f ∈ A_{p} has concentration d at degree k (measured in ℓ_{p}-norm) if

d ∥ f ∥_{p} ≤ ∥ s_{k}(f) ∥_{p}

where 0 < d ≤ 1 and k is a non-negative integer. The largest d which satisfies the above condition is called the concentration factor.

An H^{2} function f has a canonical factorization f(z) = F (z)B(z)S(z) into an outer function F, a Blaschke product B, and a singular part S. The measure M(f) of f is

M(f) = exp[∫_{0}^{2π} log |f(e^{iθ})| \frac{dθ}{2π} ].

Recall M(f) = |F (0)| ≤ ∥ F ∥_{2} = ∥ f ∥_{2} and since 1 ≤ p ≤ 2 furthermore ∥ f ∥_{2} ≤ ∥ f ∥_{p}.

For 0 ≤ λ ≤ 1, let f_{λ} : D → C be the function f_{λ}(z) = f(λz). Throughout this paper, we will assume that f(0) ≠ 0 and enumerate the zero set {z_{j}} of f so that

0 < |z_{1}| ≤ |z_{2}| ≤ … < 1.

All notation and terminology, not otherwise explained, are as in [G] or [B2].

As motivation we examine a function f ∈ A_{p} with concentration d at degree k = 0. For such a function we have that

d ∥ f ∥_{p} ≤ ∥ s_{0}(f) ∥_{p} = |f(0)| = |F (0)| S(0) ∏_{j>0} |z_{j}|.

We know that the singular part S of an H^{2} function satisfies 0 ≤ S(0) ≤ 1. Thus

d ≤ ∏_{j>0} |z_{j}|,

and since 1 − x ≤ − log x for 0 < x ≤ 1, we see that

∑_{j>0} 1 − |z_{j}| ≤ − log d .

The first bound is essentially best possible. Consider the family of H^{2} functions {f_{ε}(z) = z − ε : 0 < ε < 1}. For 1 ≤ p ≤ 2, the function f_{ε} has concentration d = \frac{ε}{1+ε} at degree 0. Thus

d ≤ ∏_{j>0} |z_{j}| = ε ≤ d(1 + ε).

We now extend these ideas for concentration at an arbitrary degree k.

2. ZEROS OF FUNCTIONS IN A_{p}

Beauzamy and Chou [BC] showed that the zeros {z_{j}} of a polynomial with concentration d at degree k (measured in ℓ_{1}-norm) satisfy

\frac{d^{k+3}}{2 9^{k+3} 3^{k^{2}}} ≤ ∏_{j>k} |z_{j}| .

Actually, their product includes the zeros outside the unit disk as well. Theorem 2.1 extends this result to functions in A_{p} and improves their constant. For our constant, we consider the function

ϕ_{d,k}(r) ≡ [\frac{(1 − r)r^{k}d}{e^{\frac{1+r}{2(1−r)}}}]^{\frac{1+r}{1−r}}

and let

Φ_{d,k} ≡ max_{0<r<1} ϕ_{d,k}(r) .

Recall that we enumerate a zero set {z_{j}} so that 0 < |z_{1}| ≤ |z_{2}| ≤ … < 1. We now fix 1 ≤ p ≤ 2.

Theorem 2.1. The zeros {z_{j}} of a function in A_{p} with concentration d at degree k (measured in ℓ_{p} - norm) satisfy

d Φ_{d,k} ≤ ∏_{j>k} |z_{j}|  and thus  ∑_{j>0} 1 − |z_{j}| ≤ k − log (d Φ_{d,k}) .

To compare with [BC], note that \frac{d^{k+3}}{2 9^{k+3} 3^{k^{2}}} < \frac{4d^{3}}{e^{2}9^{k+1}} = d ϕ_{d,k}(\frac{1}{3}) ≤ d Φ_{d,k}.

The proof uses the following lemma which extends a result of Beauzamy ([B1], [B2]).

Lemma 2.2. If a function f in A_{p} has concentration d at degree k, then

Φ_{d,k} ∥ f ∥_{p} ≤ M(f) .

As noted earlier M(f) ≤ ∥ f ∥_{p} since 1 ≤ p ≤ 2.

Proof. Let f(z) = ∑_{j≥0} a_{j}z^{j} be a function in A_{p} with appropriate concentration. Normalize so that ∥ f ∥_{p} = 1. Fix 0 < r < 1 and consider z_{0} ∈ D with |z_{0}| = r along with its Möbius function w(z) = \frac{z+z_{0}}{1+z̄_{o}z} defined on \overline{D}. Jensen’s formula and a change of variables give that

log |f(z_{0})| ≤ ∫_{0}^{2π} log |f(w(e^{iθ}))| \frac{dθ}{2π}

= ∫_{0}^{2π} log |f(e^{iθ})| \frac{1 − r^{2}}{|1 − z̄_{0}e^{iθ}|^{2}} \frac{dθ}{2π}

≤ \frac{1 − r}{1 + r} ∫_{N} log |f(e^{iθ})| \frac{dθ}{2π} + \frac{1 + r}{1 − r} ∫_{P} log |f(e^{iθ})| \frac{dθ}{2π} ,

MARIA GIRARDI 4

where

N = { θ : |f(e^{iθ})| ≤ 1} and P = { θ : |f(e^{iθ})| > 1} .

For bounding the last summand, note that

∫_{P} log |f(e^{iθ})|\frac{dθ}{2π} ≤ \frac{1}{2} ∫ |f|^{2} \frac{dθ}{2π} = \frac{1}{2} ∥ f ∥_{2}^{2} ≤ \frac{1}{2} ∥ f ∥_{p}^{2} = \frac{1}{2} .

Choose z_{0} such that |f(z_{0})| = max {|f(z)| : |z| = r}. Since

a_{j} = ∫_{0}^{2π} \frac{f(re^{iθ})}{r^{j}e^{ijθ}} \frac{dθ}{2π}  and so  |a_{j}| ≤ \frac{|f(z_{0})|}{r^{j}}

it follows that

∥ s_{k}(f) ∥_{p} ≤ ∥ s_{k}(f) ∥_{1} = ∑_{j=0}^{k} |a_{j}| ≤ |f(z_{0})| ∑_{j=0}^{k} \frac{1}{r^{j}}

≤ |f(z_{0})| r^{−k} ∑_{j=0}^{∞} r^{j} = \frac{|f(z_{0})|}{(1 − r)r^{k}} .

Combining this with the concentration information on f we see that

(1 − r)r^{k}d ≤ | f(z_{0}) | .

We now have that

log ((1 − r)r^{k}d) − \frac{1 + r}{2(1 − r)} ≤ \frac{1 − r}{1 + r} ∫_{N} log | f(e^{iθ}) | \frac{dθ}{2π} ,

which gives that ϕ_{d,k}(r) ≤ M(f), as needed.

Keeping with the previous notation, we observe a property of the measure. Lemma 2.3. If the H^{2} function f has a k-th zero of modulus λ, then

M(f_{λ}) = λ^{k}|F (0)| S(0) ∏_{j>k} |z_{j}| .

Proof. Since f_{λ} is analytic on the closed unit disk \overline{D}, Jensen’s formula provides that

M(f_{λ}) = exp ∫_{0}^{2π} log |f_{λ}(e^{iθ})| \frac{dθ}{2π} = | f_{λ}(0) | ∏_{|z_{j}|<λ} \frac{λ}{| z_{j} |} .

Now just note that |f_{λ}(0)| = |F (0)| S(0) ∏_{j>0} |z_{j}|.

This last lemma is straightforward.

Lemma 2.4. If the function f in A_{p} has concentration d at degree k and 0 < λ ≤ 1, then

(i) d ∥ f_{λ} ∥_{p} ≤ ∥ s_{k}(f_{λ}) ∥_{p}  and  (ii) λ^{k}d ∥ f ∥_{p} ≤ ∥ s_{k}(f_{λ}) ∥_{p} .

Proof. Consider f(z) = ∑_{j≥0} a_{j}z^{j} in A_{p} with appropriate concentration. Define the function h : (0, 1] → ℝ by

h(λ) = \frac{∥ f_{λ} ∥_{p}^{p}}{∥ s_{k}(f_{λ}) ∥_{p}^{p}} ≡ 1 + \frac{∑_{j=k+1}^{∞} |a_{j}|^{p}λ^{pj−pk}}{∑_{j=0}^{k} |a_{j}|^{p}λ^{pj−pk}} .

Since h is an increasing function, h(λ) ≤ h(1) ≤ d^{−p}. This provides (i). Inequality (ii) follows from the observation that

∥ s_{k}(f_{λ}) ∥_{p}^{p} = ∑_{j=0}^{k} |a_{j}|^{p} λ^{jp} ≥ ∑_{j=0}^{k} |a_{j}|^{p} λ^{kp} = λ^{kp} ∥ s_{k}(f) ∥_{p}^{p} ≥ λ^{kp}d^{p} ∥ f ∥_{p}^{p} .

The proof of Theorem 2.1 now proceeds with ease.

Proof of Theorem 2.1. Since 1 − x ≤ − log(x) for 0 < x ≤ 1, the second inequality follows from the first. Towards the first inequality, let the function f ∈ A_{p} have concentration d at degree k. We assume that f has more than k zeros for otherwise the theorem is vacuously true. Let the k^{th} zero z_{k} of f have modulus λ. The lemmas provide the following string of inequalities

Φ_{d,k} λ^{k} d ∥ f ∥_{p} ≤ Φ_{d,k} ∥ s_{k}(f_{λ}) ∥_{p}

≤ Φ_{d,k} ∥ f_{λ} ∥_{p} ≤ M(f_{λ}) = λ^{k}|F (0)| S(0) ∏_{j>k} |z_{j}| .

As noted in the introduction, S(0) ≤ 1 and |F (0)| ≤ ∥ f ∥_{p}. Thus

d Φ_{d,k} ≤ ∏_{j>k} |z_{j}| ,

as needed.

We do not believe that the bound in Theorem 2.1 is best possible. An improvement in the constant Φ_{d,k} in Lemma 2.2 would improve the bound in Theorem 2.1.

3. General Comments

In general, it is not possible to bound from below the whole product ∏_{j>0} |z_{j}| of zeros {z_{j}}_{j>0} of a function in A_{p} by a constant depending on the function’s concentration at degree k for k > 0. For example, consider the function f(z) = (z^{n} + 1)(z^{n} −ϵ) where 0 < ϵ < 1 and n is a positive integer. The whole product ∏_{j>0} |z_{j}| of zeros is ϵ yet f has concentration \frac{1}{8} (measured in ℓ_{p}-norm) at degree n. Information on the zeros of s_{k}(f) proves useful in this setting.

Theorem 3.1. If the function f in A_{p} has concentration d at degree k, then the zeros {z_{j}}_{j>0} of f and the zeros {w_{i}}_{i>0} of s_{k}(f) in the unit disk D satisfy

\frac{d}{2^{k}} ≤ \frac{∏_{j>0} |z_{j}|}{∏_{i>0} |w_{i}|} .

Thus if in the above setting s_{k}(f) has no zeros in the unit disk D, then

\frac{d}{2^{k}} ≤ ∏_{j>0} |z_{j}|.

Proof. Consider a function f in A_{p} with the appropriate concentration, i.e.

d ∥ f ∥_{p} ≤ ∥ s_{k}(f) ∥_{p} .

From Jensen’s Formula it follows [cf. M] that for the polynomial s_{k}(f) : D → C we have

\frac{∥ s_{k}(f) ∥_{p}}{2^{k}} ≤ M(s_{k}(f)) = \frac{|s_{k}(f)(0)|}{∏_{i>0} |w_{i}|} .

Basic properties of the canonical factorization f = F · B · S of f give that

\frac{|f(0)|}{∏_{j>0} |z_{j}|} = |F (0)| S(0) ≤ | F (0) | ≤ ∥ F ∥_{2} = ∥ f ∥_{2} ≤ ∥ f ∥_{p} .

The theorem now follows.

We may view any function f ∈ H^{p} as a function in the classical Lebesgue space L^{p} (T) with norm |||f|||_{p}. Beauzamy and Enflo [BE, Cor 7] showed that if the polynomials f_{1} and f_{2} in A_{2} have concentration d_{i} at degree k_{i} (respectively), then

\frac{d_{1}^{6} d_{2}^{6}}{3e^{15}9^{3(k_{1}+k_{2}+2)}} ∥ f_{1} ∥_{2} ∥ f_{2} ∥_{2} ≤ |||f_{1} f_{2}|||_{1} .

The following theorem improves on this result.

Theorem 3.2. Fix 1 ≤ p_{1}, p_{2} ≤ 2. If f_{i} ∈ A_{p_{i}} has concentration d_{i} at degree k_{i}, then

Φ_{d_{1},k_{1}} Φ_{d_{2},k_{2}} ∥ f_{1} ∥_{p_{1}} ∥ f_{2} ∥_{p_{2}} ≤ M(f_{1}f_{2}) .

Theorem 3.2 follows directly from Lemma 2.2 and the observation that M(f_{1})M(f_{2}) = M(f_{1}f_{2}). Recall that for any function g ∈ H^{2}

M(g) ≤ |||g|||_{1} .

Also

\frac{d_{1}^{6} d_{2}^{6}}{3e^{15}9^{3(k_{1}+k_{2}+2)}} < \frac{16 d_{1}^{2}d_{2}^{2}}{e^{4} 9^{k_{1}+k_{2}+2}} = ϕ_{d_{1},k_{1}} (\frac{1}{3}) ϕ_{d_{2},k_{2}} (\frac{1}{3}) ≤ Φ_{d_{1},k_{1}} Φ_{d_{2},k_{2}} .

Thus Theorem 3.2 gives (with a better constant) the result of Beauzamy and Enflo.

It is well-known that there exists positive constants k_{n}^{p} and K_{n}^{p} such that if f is a polynomial of degree n then

k_{n}^{p} ∥ f ∥_{p} ≤ M(f) ≤ K_{n}^{p} ∥ f ∥_{p} .

Lemma 2.2 gives that if f ∈ A_{p} has concentration d at degree k, then

Φ_{d,k} ∥ f ∥_{p} ≤ M(f) ≤ ∥ f ∥_{p} .

Thus Lemma 2.2 may be thought of as extending these well-known inequality from polynomials (with a certain degree) to A_{p} functions (with a certain concentration).

Of course, the settings of most interest are A_{1} and A_{2}. We worked in A_{p} with 1 ≤ p ≤ 2 as to provide unity in presentation. For p > 2, the definitions are valid but then the space A_{p} is not as natural; the ∑ |a_{j}|^{p} being finite implies that the corresponding function f(z) = ∑_{j≥0} a_{j}z^{j} is in H^{p} if and only if 1 ≤ p ≤ 2.

Our method of proof does not extend directly to the case of p > 2. Our proofs use the fact that if 1 ≤ p ≤ 2 and f is in A_{p} then M(f) ≤ ∥ f ∥_{p}. However, this basic inequality is not valid for p > 2, even in a wider sense. Specifically [cf. BN], there is a sequence of polynomials \{f_{n}\} such that M(f_{n}) ∼ \sqrt{n} and f_{n} is of degree n with coefficients of modulus 1 (thus ∥ f_{n} ∥_{p}^{p} = n + 1). For this sequence of polynomials, \frac{∥f_{n}∥_{p}}{M(f_{n})} tends to 0.

This sequence also demonstrates that, for p > 2, one does not even have the motivating bound (dependent on the concentration at degree k = 0) on the product of the zeros of a function in A_{p}. For since |f_{n}(0)| = 1, the modulus of the product of the zeros of f_{n} is [M(f_{n})]^{−1}, which grows like n^{−1/2}. However, at degree k = 0, the function f_{n} has concentration [n + 1]^{−1/p}.

References

[B1]. Bernard Beauzamy, Jensen’s inequality for polynomials with concentration at low degrees,

Numer. Math. 49 (1986), 221–225. [B2]. Bernard Beauzamy, Estimates for H^{2} functions with concentration at low degrees and

applications to complex symbolic computations, J. Reine Angew. Math. 433 (1992). [BC]. Bernard Beauzamy and Sylvia Chou, On the zeros of polynomials with concentration at

low degrees, (to appear). [BE]. Bernard Beauzamy and Per Enflo, Estimations de produits de polynômes, J. Number

Theory 21 (1985), 390–412. [BN]. E. Beller and D. J. Newman, An extremal problem for the geometric mean of polynomials,

Proc. Amer. Math. Soc. 39 (1973), 313–317. [G]. John B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [M]. K. Mahler, An application of Jensen’s formula to polynomials, Mathematika (1960), 98–

100.

current address: Department of Mathematics, University of South Carolina, Columbia, SC 29208