JENSEN TYPE INEQUALITIES AND RADIAL NULL SETS Catherine Bénéteau and Boris Korenblum

Abstract. We extend Jensen’s formula to obtain an upper estimate of log |f(0)| for analytic functions in the unit disk D that are subject to a growth restriction. Suppose we have a closed subset E of the unit circle and f in addition is continuous in the union of the open disk and E. We obtain a formula that gives an upper estimate of of log |f(0)| in terms of the values of f on E and the so-called k-entropy of E. When the set E is taken to be the whole unit circle, we get the classical Jensen’s inequality. Our formula is then applied to the study of radial null sets. 2000 Mathematics Subject Classification: 30H05, 30E25, 46E15.

1 Growth Spaces

In what follows, k denotes an increasing twice differentiable function that maps [0, 1) onto [0, ∞) and satisfies

∫_{0}^{1} k(r)dr < ∞ (1)

(2) (1 − r)k′(r)is non-decreasing

\frac{k(1 − t/2)}{k(1 − t)} ≤ C (0 < t < \frac{1}{2}) (3)

A^{<k>} denotes the Banach space of analytic functions f in D with the norm

||f||_{<k>} = sup{|f(z)| exp(−k(|z|)) : z ∈ D} < ∞.

UBA^{<k>} denotes the unit ball of A^{<k>}; it consists of f satisfying

log |f(z)| ≤ k(|z|) (z ∈ D.)

In the special case that k(r) = λ_{α}(r) = α log \frac{1}{1−r} for α > 0, we write A^{−α} for A^{<k>}.

2 Two Problems

(A) Find good (upper and lower) estimates for the quantity

J (Z, k) = sup{log |f(0)| : f ∈ UBA^{<k>}, f |_{Z}= 0}

where Z = {a_{n}} ⊂ D is a given sequence. (B) Find good estimates for

J (E, φ, k) = sup{log |f(0)| : f ∈ UBA^{<k>} ∩ C(D ∪ E), |f| |_{E}= φ}

where E ⊂ ∂D is a closed set and φ is a non-negative continuous function on E. Note that for k ≡ 0, (A^{<0>} = H^{∞}) both problems have exact solutions:

J (Z, 0) = − ∑_{n} log \frac{1}{|a_{n}|}

J (E, φ, 0) = ∫_{E} log φ(ζ)dm(ζ)

where dm is the normalized Lebesgue measure on ∂D. (Here, we assume 0 ≤ φ(ζ) ≤ 1 on E.)

3 Results for A^{−α}

Although the main thrust of this paper is problem (B), we give here for the sake of comparison the following result on problem (A) for A^{−α} (see [2] for the proof.) We define the logarithmic entropy of a finite set E ⊂ ∂D as

κ̂(E) = ∑_{n} |I_{n}| log \frac{e}{|I_{n}|}

where {I_{n}} are the complementary arcs of E and | • | denotes normalized Lebesgue measure. For a finite set S ⊂ D not containing 0, we define

T (S) = ∑\{log \frac{1}{|z|} : z ∈ S\}

and the radial projection of S :

P rS = \{\frac{z}{|z|} : z ∈ S\}.

Then we have

J (Z, λ_{α}) ≤ inf_{S⊂Z}{α[κ̂(P rS) + log κ̂(P rS)] − T (s) + α log^{+} T (s)} + C_{α}

and

J (Z, λ_{α}) ≥ inf_{S⊂Z}{α[κ̂(P rS) − log κ̂(P rS)] − T (s)} − C_{α}

where C_{α} > 0 depends only on α, and the infima are taken over all finite subsets S of Z.

\underline{COROLLARY} 3.1 For a sequence Z such that 0 is not in Z, define

D^{+}(Z) = inf{m : inf_{S⊂Z}(mκ̂(P rS) − T (s)) > −∞}.

Then D^{+}(Z) ≤ α is necessary and D^{+}(Z) < α is sufficient for Z to be an A^{−α} zero set.

Note that for other spaces A^{<k>} such that k has faster than logarithmic growth, a similar description of zero sets is not known.

4 Problem (B) for A^{<k>}

\underline{THEOREM} 4.1

J (E, φ, k) ≤ ∫_{E} max\{log φ(ζ), log p\}dm(ζ) − (log p)\frac{α}{1 − α}(1 − |E|)
+ (\frac{L}{α})^{log_{2} C}Entr_{k}(E)

where 0 < p ≤ 1 , 0 < α ≤ \frac{1}{2} are arbitrary, C is the constant in (3), L is an absolute constant, and Entr_{k}(E) is the k-entropy of E, defined as follows:

Entr_{k}(E) = ∑_{n} ∫_{1−|I_{n}|}^{1} k(t)dt

where {I_{n}} are the complementary arcs of E.

Special cases: (1) E = ∂D. Letting p → 0^{+}, we get

J (∂D, φ, k) ≤ ∫_{∂D} log φ(ζ)dm(ζ)

which is the classical Jensen’s inequality (in fact, equality.)

(2) If 0 ≤ φ(ζ) ≤ 1 on E and p = max_{ζ∈E} φ(ζ), we obtain

J (E, φ, k) ≤ (log p)\frac{|E| − α}{1 − α} + (\frac{L}{α})^{log_{2} C}Entr_{k}(E).

Choosing α = |E|/2, we get

J (E, φ, k) ≤ \frac{1}{2}(log p)|E| + (\frac{2L}{|E|})^{log_{2} C}Entr_{k}(E).

(3) If p = 1 and α = \frac{1}{2}, then

J (E, φ, k) ≤ ∫_{E} log^{+} φ(ζ)dm(ζ) + (2L)^{log_{2} C}Entr_{k}(E).

Proof: Write

∂D − E = ⋃_{n} I_{n}

where the I_{n} are open disjoint arcs on the unit circle. Call a_{n} and b_{n} the endpoints of I_{n}. Let 0 < α ≤ \frac{1}{2}. Let γ_{n} be the open arc of the circle inside the unit disk passing through a_{n} and b_{n} and forming an angle of πα (we will think of it as the normalized angle α) with the arc I_{n}. Let Γ = ⋃_{n} γ_{n}. Γ∪E forms the boundary of an open subset Ω of the unit disk containing the origin. For the proof, we construct three functions U_{1}, U_{2}, and V as follows. Step 1: Construction of U_{1} and U_{2}. Define

U_{1}(z) = ∫_{E} Re(\frac{ζ + z}{ζ − z})dm(ζ).

U_{1} is the harmonic measure of E with respect to D. \underline{LEMMA} 4.1

lim_{r→1^{−}} U_{1}(rζ) = χ_{E}(ζ) a.e. on ∂D

where χ_{E} is the characteristic function of E. In addition, U_{1}(z) ≤ α for z ∈ Γ.

Proof: The first statement is clear from the definition of U_{1} as harmonic measure. Notice that

U_{1}(z) ≤ W_{n}(z) = ∫_{∂D−I_{n}} Re(\frac{ζ + z}{ζ − z})dm(ζ)

for every n. W_{n} is the harmonic measure of ∂D − I_{n} and has a few nice geometric properties. In particular, W_{n}(z) is constant on any circle passing through a_{n} and b_{n}. In fact it is not hard to see that if z is a point in the disk, and if we consider the circle C_{n} passing through a_{n}, b_{n}, and z, then W_{n}(z) = α_{n}(z), where α_{n}(z) is the (normalized) angle between the arc I_{n} and the circle C_{n}. Therefore, for any z ∈ Γ, z ∈ γ_{n} for some n, and so U_{1}(z) ≤ W_{n}(z) ≤ α.□ Now let 0 < p ≤ 1 and define

U_{2}(z) = ∫_{E} Re(\frac{ζ + z}{ζ − z}) max\{log φ(ζ), log p\}dm(ζ).

Notice that U_{2} is harmonic in D, and

U_{2}(z) ≥ (log p)U_{1}(z) ≥ (log p)α

for z ∈ Γ, by Lemma 4.1. Step 2: Construction of V. First let K(s) = k(1 − e^{−s}) for s > 0 and extend K so that K(s) = 0 for s < 0. Now define a function

S(z) = ∑_{n} max(S_{n}^{1}(z)(1 − α_{n}(z)), S_{n}^{2}(z)(1 − α_{n}(z)))

where

S_{n}^{1}(z) = K(log |\frac{2(z − a_{n})}{(z − b_{n})(b_{n} − a_{n})}|)

and

S_{n}^{2}(z) = K(log |\frac{2(z − b_{n})}{(z − a_{n})(b_{n} − a_{n})}|).

Notice that

log |\frac{2(z − a_{n})}{(z − b_{n})(b_{n} − a_{n})}|

is harmonic in D and since (1 − r)k′(r) is non-decreasing, S_{n}^{1}(z) is subharmonic in D. Moreover, the level curves of S_{n}^{1} are orthogonal to the level curves of (1 − α_{n}(z)). Since the product of two subharmonic functions whose gradients are orthogonal is subharmonic, we conclude that S_{n}^{1}(z)(1 − α_{n}(z)) is subharmonic in D. A similar argument shows that

S_{n}^{2}(z)(1 − α_{n}(z))

is subharmonic in D. Therefore the maximum S(z) of those two functions is subharmonic in D.

\underline{LEMMA} 4.2

∫_{∂D} S(ζ)dm(ζ) ≤ Entr_{k}(E).

Proof:

∫_{∂D} S(ζ)dm(ζ) = ∫_{∂D} ∑_{n} max(S_{n}^{1}(ζ)(1 − α_{n}(ζ)), S_{n}^{2}(ζ)(1 − α_{n}(ζ)))
= ∑_{n} ∫_{I_{n}} max(S_{n}^{1}(ζ), S_{n}^{2}(ζ))dm(ζ).

Let’s study that last integral. If ζ is closer to a_{n} than to b_{n}, for example, then the integrand becomes

K(log \frac{2|ζ − b_{n}|}{|ζ − a_{n}||b_{n} − a_{n}|}) ≤ K(log \frac{2}{|ζ − a_{n}|})
≤ K(log \frac{π}{t − θ_{n}})

where ζ = e^{it}, a_{n} = e^{iθ_{n}}. A similar estimate holds when ζ is closer to b_{n} = e^{iψ_{n}}. Therefore we get:

∫_{∂D} S(ζ)dm(ζ) ≤ ∑_{n} \frac{1}{2π} (∫θ_{n}^{\frac{θn+ψn}{2}} K(log \frac{π}{t − θ_{n}})dt
+ ∫ ψ_{n}_{\frac{θn+ψn}{2}} K(log \frac{π}{ψ_{n} − t})dt)
≤ ∑_{n} ∫_{1−|I_{n}|}^{1} k(r)dr
= Entr_{k}(E).□

Assume Entr_{k}(E) is finite and define the following harmonic function

V (z) = ∫_{∂D} Re(\frac{ζ + z}{ζ − z})S(ζ)dm(ζ).

By the maximum principle, S(z) ≤ V (z) for z ∈ D.

\underline{LEMMA} 4.3 V has the following properties.

V (0) ≤ Entr_{k}(E)
V (ζ) = 0 for ζ ∈ E
V (z) ≥ (\frac{L}{α})^{− log_{2} C}k(|z|) for some absolute constant L and for z ∈ Γ.

Proof: The first two properties are immediate from the definition of V and by lemma 4.2. For the third, let us examine the behavior of V on Γ. First of all, it is geometrically clear that for z ∈ Γ, there exists an absolute constant L such that if z ∈ γ_{n},

0 < \frac{min(|z − a_{n}|, |z − b_{n}|)}{1 − |z|} ≤ \frac{L}{α} < ∞.

Therefore for z ∈ γ_{n}, (let’s say z is closer to a_{n} than to b_{n})

V (z) ≥ S(z)
≥ K(log \frac{2|z − b_{n}|}{|z − a_{n}||b_{n} − a_{n}|})
≥ K(log \frac{1}{|z − a_{n}|})
≥ K(log \frac{α}{L(1 − |z|)})
= k(1 − \frac{L}{α}(1 − |z|))
≥ C^{[log_{2} \frac{L}{α}]+1}k(1 − (1 − |z|))

(by property (3) of k)

≥ C^{log^{2} ^{\frac{2L}{α}}} k(|z|)
= (\frac{2L}{α})^{log_{2} C}k(|z|).

By relabeling L, we get the statement of the lemma.□ Step 3: Construction of H and application of the maximum principle. Finally, let us define

H(z) = U_{2}(z) − (log p)\frac{α}{1 − α}(1 − U_{1}(z)) + (\frac{L}{α})^{− log_{2} C}V (z).

H is harmonic in the disk. Moreover, for ζ ∈ E,

H(ζ) ≥ log φ(ζ) = log |f(ζ)|.

On the other hand, if z ∈ Γ,

H(z) ≥ (log p)α − (log p)\frac{α}{1 − α}(1 − α) + k(|z|) = k(|z|) ≥ log |f(z)|.

Recall that Ω is the part of the unit disk that is bounded by Γ ∪ E. We therefore have a harmonic function H whose values on the boundary of Ω dominate the boundary values of log |f(z)|, a function that is subharmonic in Ω. By the maximum principle, we can conclude that

log |f(0)| ≤ H(0)
≤ ∫_{E} max\{log φ(ζ), log p\}dm(ζ) − (log p)\frac{α}{1 − α}(1 − |E|)
+ (\frac{L}{α})^{log_{2} C}Entr_{k}(E)

as desired.□

\underline{COROLLARY} 4.1 If f ∈ A^{<k>} and

lim_{r→1^{−}} f(rζ) = 0

uniformly in ζ ∈ E, and if |E| > 0 and Entr_{k}(E) < ∞, then f ≡ 0.

Remark: It is well-known (Lusin-Privalov theorem) that there are non-zero analytic functions that have zero radial limits on a set E of full Lebesgue measure [4]. (However, this cannot happen if |E| > 0 and at the same time E is of the second Baire category.) More specifically, no matter how slowly k(r) tends to +∞, there is always a non-zero f in A^{<k>} such that lim_{r→1^{−}} f(rζ) = 0 a.e. [3]. By Egoroff’s theorem, f may have uniform radial limits 0 on a closed set E whose measure is arbitrarily close to the full measure of ∂D. Corollary 4.1 shows that such uniform radial null sets E with |E| > 0 must have infinite k-entropy. A similar phenomenon was discovered by S. V. Hrus̆c̆ev (see [1], p. 278-305) in connection with the Khinchin-Ostrowski property.

Acknowledgement. The authors thank the referee for careful reading of the manuscript and useful suggestions.

References.

1. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer-Verlag 1994.

2. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer-Verlag 2000.

3. Kahane, J.P., Katznelson, Y.: Sur le comportement radial des fonctions analytiques. C. R. Acad. Sc. Paris, Série A, 272, 718-719 (1971).

4. Privalov, I.I.: Randeigenschaften analytischer funktionen. Deutscher, V. der Wiss., Berlin 1956. (German translation of the Russian book, Moscow 1950.)

Catherine Bénéteau Department of Mathematics Seton Hall University 400 South Orange Avenue South Orange NJ 07079 E-mail: beneteca@shu.edu

Boris Korenblum Department of Mathematics University at Albany Albany NY 12222