Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 305–312

ON COEFFICIENTS, BOUNDARY SIZE AND HÖLDER DOMAINS

Nikolai G. Makarov and Christian Pommerenke

Caltech, Department of Mathematics, 253-37 Pasadena, CA 91125, U.S.A.; makarov@cco.caltech.edu Technische Universität, Fachbereich Mathematik D-10623 Berlin, Germany; pommeren@math.tu-berlin.de

Abstract. Let ϕ(z) = ∑ a_{n}z^{n} map the unit disk onto a bounded plane domain. Then (∗) a_{n} = O(n^{γ−1+ε}) for every ε > 0, where γ is an unknown absolute constant. We show here that γ < 0.4886. Suppose now additionally that the boundary of the image domain has Minkowski dimension ≤ M. It is shown that still no bound better than (∗) holds provided that M_{c} ≤ M ≤ 2 for some critical dimension M_{c} < 2. The proof is based on the universal integral means spectrum, the connection with Hölder continuity and the Carleson–Jones modification method.

1. Coefficients and boundary size

Let mdim denote the upper Minkowski (or box counting) dimension. In its definition, in contrast to the Hausdorff dimension, only coverings by disks of the same size are allowed. For 1 ≤ M ≤ 2, we denote by S^{(M)} the class of bounded (injective) conformal maps

φ(z) = ∑_{n=0}^{∞} a_{n}z^{n}

of the unit disk D such that

(1.1)  mdim ∂φ(D) ≤ M.

In particular S^{(2)} consists of all bounded conformal maps.

For 1 ≤ M ≤ 2, we define

(1.2)  γ^{(M)} = sup\{lim sup_{n→∞} \frac{log(n|a_{n}|)}{log n} : φ ∈ S^{(M)}\}.

We write γ = γ^{(2)} . Thus γ is the smallest exponent such that

(1.3)  a_{n} = O(n^{γ+ε−1}), (n → ∞), for every ε > 0,

1991 Mathematics Subject Classification: Primary 30C50.

for all bounded univalent functions or, equivalently [CJ92], for all univalent functions φ(z) = z + a_{0} + a_{1}z^{−1} + ⋯ in {|z| > 1}. At present the best lower bounds are

(1.4)  \{ \begin{matrix}γ > 0.17, & analytic  [P67], \\ γ > 0.24, & experimental [CJ92], see also [K96]. \end{matrix}

Carleson and Jones have conjectured that γ = \frac{1}{4} .

The upper bound γ < 0.4910 [CP67], [P92, p. 183] will be slightly improved below using a method of Carleson and Jones [CJ92].

Theorem. We have γ < 0.4886. There exists a constant M_{c} with

(1.5)  1.2048 < \frac{1}{1 − γ} < M_{c} < 1 + \frac{1}{1 + γ^{2}/(8π)} < 1.9989

such that

(1.6)  γ^{(M)} = γ for M_{c} ≤ M ≤ 2,

(1.7)  γ^{(M)} < γ for 1 ≤ M < M_{c}.

Furthermore, γ^{(1+δ)} = δ − δ^{2} + O(δ^{3}) as δ → 0.

The critical dimension M_{c} is given by

(1.8)  M_{c} = 1 + \frac{B(1)}{1 − B′(1+)},

where B(t) is the universal integral means spectrum defined in (2.2) below. The theorem shows that there is a phase transition at M_{c}: for dimensions M ≥ M_{c} the additional information that mdim ∂φ(D) ≤ M becomes irrelevant for the coefficient problem.

Hölder domains will play an essential role in our proof; see Proposition 2. We want to thank Steffen Rohde for our discussions.

2. The integral means spectrum and Hölder domains

The integral means spectrum of a conformal map φ of D into C is defined by

(2.1)  β_{φ}(t) = lim sup_{r→1−} \frac{log ∫_{T} |φ′(rζ)|^{t} |dζ|}{| log(1 − r)|}, −∞ < t < +∞,

where T = ∂D. The universal integral means spectrum, defined by

(2.2)  B(t) = sup{β_{φ}(t) : φ bounded conformal map},

determines many growth properties. It is elementary that B(t) is convex and that B(t) = t − 1 for t ≥ 2. It is furthermore known that

(2.3)  B(t) = |t| − 1 for t ≤ t_{0}, [CM94],
(2.4) c_{1}t^{2} ≤ B(t) ≤ c_{2}t^{2} for |t| ≤ c_{3}, [M86],[CP67],
(2.5)  B(t) = t − 1 + O((t − 2)^{2}) as t → 2, [JM95, Thm. D].

The constant t_{0} ≤ −2 in (2.3) is unknown and Brennan [B78] has conjectured that t_{0} = −2. See [P92, p. 178] for quantitative estimates.

We need the following characterization of the integral means spectra possible for bounded univalent functions. See [M95, Theorem 5.2] for the proof by fractal approximation.

Proposition 1. Let β: R → [0, +∞) be convex. Then β = β_{φ} for some φ ∈ S^{(2)} if and only if

(1) 0 ≤ β(t) ≤ B(t), (t ∈ R),

(2) |t|β′(t±) ≤ 1 + β(t), (t ∈ R).

The condition (2) means that the tangents to the graph y = β(t) intersect the y -axis in the interval [−1, 0].

For 0 < η ≤ 1, let S_{η} denote the class of conformal maps φ such that

(2.6)  φ′(z) = O((1 − |z|)^{η−1}) as |z| → 1.

This Hölder class S_{η} is much easier to handle than S^{(M)} , for instance because whether φ ∈ S_{η} is essentially determined by β′_{φ}(+∞). We define

(2.7)  B_{η}(t) = sup{β_{φ}(t) : φ ∈ S_{η}}, (t ∈ R).

Smith and Stegenga [SS91] have shown that B_{η}(2) < 2.

Proposition 2. Consider the tangent

(2.8)  T_{η}(t) = (1 − η)t + min_{τ}(B(τ) − (1 − η)τ)

of slope 1 − η to the graph y = B(t) and define t_{η} by T_{η}(t_{η}) = B(t_{η}). Then

(2.9)  B_{η}(t) = \{ \begin{matrix}B(t), & for −∞ < t ≤ t_{η}, \\ T_{η}(t), & for t_{η} ≤ t < +∞. \end{matrix}

Proof. Let B_{η}^{∗}(t) denote the right-hand side of (2.9). If φ ∈ S_{η}, it follows from (2.6) and (2.1) that

β_{φ}(t) ≤ min{β_{φ}(τ) + (1 − η)(t − τ) : τ ≤ t},

so that B_{η}(t) ≤ B_{η}^{∗}(t) by (2.8).

308 Nikolai G. Makarov and Christian Pommerenke

Conversely let ε > 0 be given. We may assume that η < 1. By Proposition 1 there exists φ ∈ S^{(2)} such that β_{φ}(t) = (1 − ε)B_{η}^{∗}(t). The Koebe distortion theorem shows that c_{1}|φ′(z)| ≤ |φ′(ze^{iθ})| for |z| = r < 1, |θ| ≤ 1 − r. Hence, for t > 0,

(1 − r)c_{1}^{t} max_{|z|=r} |φ′(z)|^{t} ≤ ∫_{T} |φ′(rζ)|^{t}| dζ| < (\frac{1}{1 − r} )^{β_{φ}(t)+1}

for r > r_{0}(t). Since B_{η}^{∗}(t) = c_{2} + (1 − η)t for large t, it follows that

max_{|z|=r} |φ′(z)| ≤ c_{3}(\frac{1}{1 − r} )^{(1−ε)(1−η)+c_{4}/t} < c_{3}(\frac{1}{1 − r} )^{1−η}

and thus φ ∈ S_{η}. Hence B_{η}(t) ≥ (1−ε)B_{η}^{∗}(t) for every ε > 0 which implies (2.9).

Proposition 3. (i) If φ ∈ S^{(M)} then

(2.10)  β_{φ}(M) ≤ M − 1.

(ii) If (2.10) holds and if moreover φ(D) is a Hölder domain then φ ∈ S^{(M)} .

See [P92, p. 241] for the proof of (i) and [M95, (3.3)] for the proof of (ii). Note that (ii) is not true for arbitrary bounded domains [P92, p. 241].

3. The Carleson–Jones modification We also need the following result [CJ92] that Carleson and Jones used to

prove

(3.1)  γ ≡ γ^{(2)} = B(1).

Proposition 4. Let φ(z) = ∑_{k=0}^{∞} a_{k}z^{k} be univalent in D and consider the Fejér-type means

(3.2)  p_{n}(z) = \frac{1}{2π} ∫_{−π}^{π} ∑_{ν=0}^{2n}(1 − \frac{|ν − n|}{n + 1})z^{ν}e^{i(n−ν)θ} \frac{|φ′(r_{n}e^{iθ})|}{φ′(r_{n}e^{iθ})} dθ,

where r_{n} = 1 − 1600/n, n > 1600. If 0 < δ ≤ δ_{0} , then

(3.3)  φ_{n}(z) = φ(r_{n}z) + \frac{δ}{n}p_{n}(z)φ′(r_{n}z) = ∑_{k=0}^{∞} a_{nk}z^{k}

is univalent in D and satisfies

(3.4)  |a_{nn}| ≥ \frac{δ}{4πn} ∫_{−π}^{π} |φ′(r_{n}e^{iθ})| dθ − |a_{n}|.

If moreover |φ′(z)| ≤ A(1 − |z|)^{η−1} , then

(3.5)  |φ′_{n}(z)| ≤ (1 + 3δ)A(1 − |z|)^{η−1} for z ∈ D.

Only the final statement is not formulated in [CJ92]. The polynomial p_{n} has degree 2n and satisfies |p_{n}(z)| ≤ 1 for |z| ≤ 1 by properties of the Fejér kernel. Thus, Bernstein’s inequality [Z68, p. II.11] implies that |p′_{n}(z)| ≤ 2n. Hence by (3.3),

\begin{aligned}|φ′_{n}(z)| & = |r_{n}φ′(r_{n}z) + \frac{δ}{n}p′_{n}(z)φ′(r_{n}z) + \frac{δ}{n}p_{n}(z)r_{n}φ′′(r_{n}z)| \\ & ≤ (1 + 2δ)|φ′(r_{n}z)| + \frac{δ}{n}|φ′′(r_{n}z)|. \end{aligned}

The distortion theorem shows that

|φ′′(r_{n}z)| ≤ \frac{6}{1 − r_{n}} |φ′(r_{n}z)| < n|φ′(r_{n}z)|

and (3.5) follows from our assumption.

An immediate consequence is that

(3.6)  B_{η}(1) = γ_{η} ≡ sup\{lim sup_{n→∞} \frac{log(n|a_{n}|)}{log n} : φ ∈ S_{η}\};

see (3.4), (3.5) and the trivial estimate

(3.7)  |a_{n}|r_{n}^{n−1} ≤ \frac{1}{2π} ∫_{T} |f′(r_{n}ζ)| |dζ|.

4. Estimates of integral means

The next estimates are easy consequences of known results, in particular of Carleson and Jones [CJ92]. Let φ map D conformally onto a bounded domain. We write

(4.1)  A(r, α) = {ζ ∈ T : |φ′(rζ)| < (1 − r)^{−1+α}}, (0 < r < 1, α > 0),

and we assume that

(4.2)  ∫_{A(r,α)} |φ′(rζ)|^{2}| dζ| = O((1 − r)^{−1+g(α)}), (r → 1),

where g(α) is a continuous function depending only on α. Carleson and Jones have shown that one can choose

(4.3)  g(α) = α^{3}/(8π);

this is implicit in [CJ92, p. 204–205]. Their proof uses a Marcinkiewicz integral; see also [M95, (4.3)].

While (4.2) is useful for estimating β_{φ}(t) for t near to 2, the estimate [CP67], [P92, p. 178]

(4.4)  β_{φ}(t) ≤ t − \frac{1}{2} + (4t^{2} − t + \frac{1}{4})^{1/2} ≤ 3t^{2} + 7t^{3}

is useful for small t. Let γ again be defined by (1.3); compare (3.1).

Proposition 5. Let (4.2) be satisfied for all bounded univalent functions φ. Then

(4.5)  B(1) = γ ≤ max[α, 0.4908 − 0.482g(α)]

for each α > 0 and

(4.6)  B(t) ≤ max[α + (1 − α)(t − 1), γ + (1 − γ − g(α))(t − 1)]

for 1 ≤ t ≤ 2, α > 0. Furthermore,

(4.7)  γ < B′(1+) ≤ 1 − γ − g(γ).

Proof. It follows from (4.1) that, for 0 < t < 2,

(4.8)  \begin{aligned}∫_{T\A(r,α)} |φ′(rζ)|^{t} |dζ| & ≤ (1 − r)^{(1−α)(2−t)} ∫_{T} |φ′(rζ)|^{2} |dζ| \\ & = O((1 − r)^{(1−α)(2−t)−1}) \end{aligned}

because φ(D) is bounded and thus has finite area.

Writing 1 = a + (1 − a) with 0 < a < 1, we see from Hölder’s inequality that

∫_{A(r,α)} |φ′(rζ)| |dζ| ≤ (∫_{T} |φ′|^{2a/(1+a)}| dζ|)^{(1+a)/2}(∫_{A(r,α)} |φ′|^{2} |dζ|)^{(1−a)/2}.

We estimate the first factor by (4.4) and the second factor by (4.2) and obtain the bound O((1 − r)^{−δ}), where

δ = \frac{6a^{2}}{1 + a} + \frac{28a^{3}}{(1 + a)^{2}} + (1 − g(α))\frac{1 − a}{2},

and (4.5) follows from (4.8) (with t = 1) if we choose a = 0.036.

Writing t = (2 − t) + 2(t − 1) with 1 < t < 2, we also see from Hölder’s inequality that

∫_{A(r,α)} |φ′(rζ)|^{t} |dζ| ≤ (∫_{T} |φ′| |dζ|)^{2−t}(∫_{A(r,α)} |φ′|^{2} |dζ|)^{t−1}.

Using (3.1) and (4.2) we obtain the bound O((1 − r)^{−δ}) with

δ = (γ + ε)(2 − t) + (1 − g(α))(t − 1)

for every ε > 0, and (4.6) now follows from (4.8).

Finally, if 0 < α < γ, then by (4.6) we have

\frac{B(t) − B(1)}{t − 1} ≤ max[\frac{α − γ}{t − 1} + 1 − α, 1 − γ − g(α)] = 1 − γ − g(α)

for small t−1 > 0. Hence B′(1+) ≤ 1−γ−g(α) for α < γ and thus ≤ 1−γ−g(γ). The estimate B′(1+) > γ follows by convexity from B(0) = 0 and B(1) = γ.

5. Proof of the theorem

(a) If follows from (4.5) and the Carleson–Jones estimate (4.3) that

γ ≤ max[α, 0.4908 − 0.482α^{3}/(8π)] < 0.4886

if we choose α slightly smaller than 0.4886.

(b) Let M_{c} be defined by (1.8). It follows from (4.7) and the Carleson–Jones estimate (4.3) that

1 − γ > 1 − B′(1+) ≥ γ + γ^{3}/(8π),

and (1.5) follows from (1.8) and the bound γ > 0.17 in (1.4); the experimental bound γ > 0.24 would give 1.3157 < M_{c} < 1.9978.

(c) Let 1 ≤ M < 2. The tangent from the point (M, M − 1) to the graph y = B(t) is given by (2.8), where η > 0 is chosen such that

(5.1)  min_{τ}[B(τ) + (1 − η)(M − τ)] = M − 1.

The point t_{η} determined by T_{η}(t_{η}) = B(t_{η}) satisfies t_{η} ≤ M. If φ ∈ S_{η} then β_{φ}(M) ≤ B_{η}(M) = T_{η}(M) = M − 1 by Proposition 2, and thus φ ∈ S^{(M)} by Proposition 3(ii). Hence

(5.2)  B_{η}(1) = γ_{η} ≤ γ^{(M)} ≤ B^{(M)}(1)

by (3.6), (1.2) and (3.7).

Assume first that t_{η} ≥ 1. Then

(5.3)  B^{(M)}(1) ≤ B(1) = γ = B_{η}(1)

by (2.9). Hence equality holds everywhere in (5.2) and (5.3), in particular we have γ^{(M)} = γ.

Assume now that t_{η} < 1. Since B^{(M)}(t_{η}) ≤ B(t_{η}) = T_{η}(t_{η}) and also B^{(M)}(M) ≤ M − 1 = T_{η}(M) by Proposition 3(i) and (5.1), it follows from the convexity of B^{(M)} that

(5.4)  B^{(M)}(1) ≤ T_{η}(1) = B_{η}(1).

Hence we have equality everywhere in (5.2) and (5.4) so that γ^{(M)} = T_{η}(1) < B(1) = γ.

The phase transition occurs when t_{η} = 1. In this case, the tangent T (t) = B′(1+)(t − 1) + B(1) to the graph y = B(t) at 1 intersects the line y = t − 1 at the point t = M_{c} given by (1.8).

312 Nikolai G. Makarov and Christian Pommerenke

(d) Finally let M = 1 + δ and δ → 0+. The tangent is

(5.5)  T_{η}(t) = B(t_{η}) + B′(t_{η}+)(t − t_{η}),

where t_{η} is given by

(5.6)  δ = T_{η}(1 + δ) = B(t_{η}) + B′(t_{η}+)(1 + δ − t_{η}).

Since B′(t_{η}+) > c_{1}t by (2.4), it follows first that t_{η} = O(δ) and next that B′(t_{η}+) = δ + O(δ^{2}). Hence (5.5) and (5.6) imply that

γ^{(1+δ)} = T_{η}(1) = δ(1 − B′(t_{η}+)) = δ − δ^{2} + O(δ^{3}).

References

[B78] Brennan, J.E.: The integrability of the derivative in conformal mapping. - J. London Math. Soc. (2) 18, 1978, 261–272.

[CJ92] Carleson, L., and P.W. Jones: On coefficient problems for univalent functions and conformal dimension. - Duke Math. J. 66, 1992, 169–206.

[CM94] Carleson, L., and N.G. Makarov: Some results connected with Brennan’s conjecture. - Ark. Mat. 32, 1994, 33–62.

[CP67] Clunie, J., and Ch. Pommerenke: On the coefficients of univalent functions. - Michigan Math. J. 14, 1967, 71–78.

[K96] Kraetzer, Ph.: Experimental bounds for the integral means spectrum of conformal maps. - Complex Variables Theory Appl. 31, 1996, 305–309.

[JM95] Jones, P.W., and N.G. Makarov: Density properties of harmonic measure. - Ann. Math. 142, 1995, 427–455.

[M86] Makarov, N.G.: A note on the integral means of the derivative in conformal mapping. - Proc. Amer. Math. Soc. 96, 1986, 233–236.

[M95] Makarov, N.G.: Fine structure of harmonic measure. - Recent Advances in Analysis, Conference in Strobl 1994, London Math. Soc. Lecture Series (to appear).

[P67] Pommerenke, Ch.: Relations between the coefficients of a univalent function. - Invent. Math. 3, 1967, 1–15.

[P92] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. - Springer-Verlag, 1992.

[SS91] Smith, W., and D.A. Stegenga: Exponential integrability of the quasihyperbolic metric on Hölder domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 1991, 345–360.

[Z68] Zygmund, A.: Trigonometric Series. - Cambridge Univ. Press, 1968.

Received 9 October 1995