Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 357–364

THE LAW OF THE ITERATED LOGARITHM FOR LOCALLY UNIVALENT FUNCTIONS

I. R. Kayumov

Kazan State University, Chebotarev Institute of Mathematics and Mechanics Universitetskaya 17, Kazan 420008, Russia; ikayumov@ksu.ru

Abstract. In this paper we prove a sharp version of the Makarov law of the iterated logarithm. In particular, we show that the constant in the right side of this law depends on an asymptotic behaviour of the integral means of the derivative of an analytic function. Also, we establish that this constant is equal to the asymptotic variance for some domains with fractal type boundaries.

Let f be an analytic and univalent function in the unit disk D = {|z| < 1}. Makarov [5] proved that there exists a universal constant C > 0 such that

(1)  lim sup_{r→1−} \frac{| log f′(rζ)|}{\sqrt{log(1/(1 − r)) log log log(1/(1 − r))}} ≤ C∥ log f′∥_{B}

for almost all ζ on |ζ| = 1, where

∥ log f′∥_{B} = | log f′(0)| + sup_{|z|<1}(1 − |z|^{2})|\frac{f′′}{f′} (z)|

is the Bloch norm. Pommerenke [8, p. 186] showed that this inequality is true for C = 1 and there is a univalent function for which the inequality is false for C ≤ 0.685. Therefore, this result is not far from being the best possible. Przytycki, Urbański and Zdunik [9] established that for some classes of domains with fractal type boundaries the equality holds with \sqrt{σ^{2}} in the right side of (1) where

σ^{2} = \frac{1}{2π} lim sup_{r→1} \frac{∫ | log f′|^{2} dθ}{log(1/(1 − r))}

is the asymptotic variance. In the paper [9] the authors used another definition of the asymptotic variance. But, in fact, their definition is equal to our definition in the “fractal” case.

2000 Mathematics Subject Classification: Primary 30C55.

The goal of this paper is to obtain a sharp version of the Makarov law of the iterated logarithm for locally univalent functions, i.e. for functions f for which f′(z) ≠ 0, z ∈ D.

Let f be a locally univalent function in the unit disk D and p be a complex number. Then for all δ > 0 we define

β_{δ}(p) = sup_{r∈[0,1)} \frac{log[δ ∫_{|z|=r} |f′(z)^{p}|dθ]}{log(1/(1 − r))}.

In other words β_{δ}(p) is the minimal number for which

∫ |f^{′p}| dθ ≤ \frac{1}{δ}(\frac{1}{1 − r} )^{β_{δ}(p)}, 0 ≤ r < 1.

If p is a real number then

β_{δ}(p) → β(p) as δ → 0,

where

β(p) = lim sup_{r→1} \frac{log ∫_{|z|=r} |f′(z)|^{p} dθ}{log(1/(1 − r))}

is the classical integral means spectrum [8, p. 176].

It follows from the integral means spectrum concept ( [2], [7], [8]) that there is a connection between geometric properties of domains and the integral means spectrum. On the other hand, Makarov [5] established that the law of the iterated logarithm is closely related to the boundary properties of conformal maps. This leads to the following natural question: Is there a simple relation between the law of the iterated logarithm and the integral means spectrum? A possible answer for this question is the following result which we will prove later.

Suppose f is a locally univalent function in the unit disk and δ > 0. Then

lim sup_{r→1−} \frac{| log f′(rζ)|}{\sqrt{log(1/(1 − r)) log log log(1/(1 − r))}} ≤ 2 lim sup_{p→0} \frac{\sqrt{β_{δ}(p)}}{|p|}

for almost all ζ on |ζ| = 1.

It is more convenient for us to formulate this result in the form of the following

Theorem 1. Suppose f is a locally univalent function in the unit disk. Then the following inequality holds

(2)  lim sup_{r→1−} \frac{| log f′(rζ)|}{\sqrt{log(1/(1 − r)) log log log(1/(1 − r))}} ≤ \sqrt{σ^{2}(0+)}

for almost all ζ on |ζ| = 1, where

σ^{2}(δ) = 4 lim sup_{p→0} \frac{β_{δ}(p)}{|p|^{2}}.

We remark that Pommerenke’s result with the constant C = 1 easily follows from our theorem because it is known [3] that if log f′ is a Bloch function then σ^{2}(0+) ≤ ∥ log f′∥_{B}^{2} . Moreover, we will see that in many cases

σ^{2}(0+) = σ^{2}.

Further, it is convenient to use the following abbreviation:

∫_{0}^{2π} h(re^{iθ}) dθ ≡ ∫ h dθ

The next lemma can be deduced from Makarov’s proof of the law of the iterated logarithm [5].

Lemma 1. Let C_{k} be a sequence of positive numbers and C_{k}^{1/k} → 1 as k → ∞. If

∫ | log f′|^{2n} dθ ≤ C_{n}n!A^{2n} log^{n} \frac{1}{1 − r}

for all natural n and for all r ∈ [1 − exp(− exp e^{n}), 1), then

lim sup_{r→1−} \frac{| log f′(rζ)|}{\sqrt{log(1/(1 − r)) log log log(1/(1 − r))}} ≤ A

for almost all ζ on |ζ| = 1.

Proof. We use Pommerenke’s version of Makarov’s law [8, p. 186]. Our proof is almost the same as Pommerenke’s proof. Let us only remark that instead of ∫_{e}^{∞} in his proof we have to consider ∫_{exp(e^{n})}^{∞}.

Proof of Theorem 1. Fix δ > 0 and ε > 0. Then there exists p_{0} = p_{0}(δ, ε) > 0 such that

∫ |f^{′p}| dθ ≤ \frac{1}{δ}(\frac{1}{1 − r} )^{(σ^{2}(δ)+ε)|p|^{2}/4}

for |p| < p_{0} . This implies

\begin{aligned}∫ I_{0}(t| log f′|) & = ∫ \frac{1}{2πt} ∫_{|p|=t} |f^{′p}| |dp| dθ = \frac{1}{2πt} ∫_{|p|=t} ∫ |f^{′p}| d θ|dp| \\ & ≤ \frac{1}{δ}(\frac{1}{1 − r} )^{(σ^{2}(δ)+ε)t^{2}/4}, t ∈ (0, p_{0}), \end{aligned}

where

I_{0}(x) = ∑_{k=0}^{∞}(\frac{x^{2}}{4} )^{k}/k!^{2} = \frac{1}{2π} ∫_{0}^{2π} e^{x cos θ} dθ

is the modified Bessel function of order zero [1]. At the same time,

∫ | log f′|^{2n} dθ ≤ \frac{4^{n}}{t^{2n}} n!^{2} ∫ I_{0}(t| log f′|) dθ ≤ \frac{4^{n}}{t^{2n}} n!^{2} \frac{1}{δ}(\frac{1}{1 − r} )^{(σ^{2}(δ)+ε)t^{2}/4}.

Setting

t^{2} = \frac{4n}{(σ^{2}(δ) + ε) log(1/(1 − r))}, n ≤ log log log \frac{1}{1 − r} ,

and using the identity

e = (\frac{1}{1 − r} )^{1/ log(1/(1−r))}

we obtain

∫ | log f′|^{2n} dθ ≤ \frac{1}{δ}n!^{2}e^{n} \frac{1}{n^{n}} (σ^{2}(δ) + ε)^{n}(log \frac{1}{1 − r} )^{n}.

Applying Lemma 1, we get

lim sup_{r→1−} \frac{| log f′(rζ)|}{\sqrt{log(1/(1 − r)) log log log(1/(1 − r))}} ≤ \sqrt{σ^{2}(δ) + ε}

for almost all ζ on |ζ| = 1 and for all δ > 0, ε > 0. Hence, this result is true for δ = 0+ and ε = 0. This completes the proof.

If f is a univalent function and f(D) is a domain with rectifiable boundary then inequality (2) is trivially sharp. Non-trivial examples, which show that this inequality is sharp, can be obtained by using lacunary series.

Let log f′ = ∑_{k=1}^{∞} a_{k}z^{n_{k}} be a lacunary series with bounded coefficients and n_{k+1}/n_{k} ≥ q > 1. Since log f′ is a Bloch function then σ^{2}(0+) < +∞ as was mentioned above. In the other direction, Makarov [6] showed that if n_{k} = 2^{k} and a_{k} = 1 for all k then σ^{2}(0+) > 0. Rohde [8] improved his result in the following sense. Suppose q is an integer, n_{k} = q^{k} and a_{k} = a > 0, then σ^{2}(0+) ≥ a^{2}/ log q.

In [4] it was shown that if q ≥ 2 then

σ^{2}(0+) = lim sup_{r→1} \frac{B^{2}}{log(1/(1 − r))},

where B^{2} = ∑_{k=1}^{∞} |a_{k}|^{2}r^{2n_{k}}.

We want to extend this result for the case q > 1. To do this we need the following

Definition ([10]). We say that a lacunary series satisfies condition (ϱ, R) if it consists of blocks of terms of length R, separated by empty blocks of length ϱ.

Weiss [10] proved the following

Lemma 2. Let f(z) = ∑_{k=1}^{∞} a_{k}z^{n_{k}} be a lacunary series satisfying condition (ϱ, R), where

(q^{ϱ/3} − 1)^{−1} ≤ \frac{1}{4}(q − 1)  and  q^{−R/3}(R + 1)^{2} ≤ \frac{1}{2}.

Let M = sup |a_{k}|, B^{2} = ∑_{n=1}^{∞} |a_{k}|^{2}r^{2n_{k}}, r ∈ [0, 1). Then

πe^{(1−ctR^{2})t^{2}B^{2}/4} ≤ ∫ e^{t Re f(re^{iθ})} dθ ≤ 3πe^{(1+ctR^{2})t^{2}B^{2}/4}.

Denote by ϱ_{0} the minimal positive number for which

(q^{ϱ_{0}/3} − 1)^{−1} ≤ \frac{1}{4}(q − 1)  and  q^{−ϱ_{0}/3}(ϱ_{0} + 1)^{2} ≤ \frac{1}{2}.

Now, we can prove the following

Theorem 2. Let log f′ = ∑_{k=1}^{∞} a_{k}z^{n_{k}} be a lacunary series with bounded coefficiens for which n_{k+1}/n_{k} ≥ q > 1. Then

σ^{2}(0+) = lim sup_{r→1} \frac{B^{2}}{log(1/(1 − r))},

where B^{2} = ∑_{k=1}^{∞} |a_{k}|^{2}r^{2n_{k}}.

Proof. It is clear that we can represent log f′ as f_{1} + f_{2} , where f_{1} satisfies the (ϱ_{0}, R)-condition and f_{2} satisfies the (R, ϱ_{0})-condition. Let us estimate B_{1}^{2} and B_{2}^{2} . In fact, it is enough to estimate only B_{2} because B^{2} = B_{1}^{2} + B_{2}^{2} . We have

B_{2} ≤ M^{2} ∑_{j=1}^{∞} ∑_{k=jR+(j−1)ϱ_{0}}^{j(R+ϱ_{0})} r^{2q^{k}} ≤ M^{2}ϱ_{0} ∑_{j=1}^{∞} r^{2q^{Rj}} ≤ C log \frac{1}{1 − r} /R.

Evidently, without loss of generality, we can assume that p = t is a real positive number. Setting R = t^{−2/5} , we have

∫ |f′|^{t} dθ = ∫ e^{t Re f_{1}+t Re f_{2}} dθ ≤ (∫ e^{αt Re f_{1}} dθ)^{1/α}(∫ e^{βt Re f_{2}} dθ)^{1/β},

where β = t^{−1/5} and α = (1 − t^{1/5})^{−1} . Applying Lemma 2 with f_{1} and f_{2} and the estimate for B_{2} we obtain

∫ |f′|^{t} dθ ≤ Ce^{B^{2}t^{2}/4}(\frac{1}{1 − r} )^{Ct^{2+1/5}}.

Analogously,

∫ |f′|^{t} dθ ≥ Ce^{B^{2}t^{2}/4}(1 − r)^{Ct^{2+1/5}}.

This concludes the proof.

Applying the law of the iterated logarithm for lacunary series [10] it is easy to show that if

lim_{r→1} \frac{B^{2}}{log(1/(1 − r))}

exists then the equality in (2) holds. Other examples which show that (2) is sharp come from the theory of Julia sets. The idea for using Julia sets in the theory of univalent functions is due to Carleson and Jones [2]. They conjectured that the basin of attraction of infinity for an iteration z^{2} + c for some c maximizes β_{0+}(1) in the class Σ.

Let F (z) = z^{q} + a_{q−1}z^{q−1} + ⋯ be a polynomial of degree q ≥ 2 and

Ω = {ζ : F^{on}(ζ) → ∞ as n → ∞}

be the basin of attraction of ∞ for F.

Theorem 3. Let Ω be a simply connected John domain. Then

σ^{2}(0+) = σ^{2} = lim sup_{r→1} \frac{B^{2}}{log(1/(1 − r))},

where B^{2} = ∑ |a_{k}|^{2}r^{k}; a_{k} are the coefficients of log f′ and f is the conformal mapping from D^{−} = {|ζ| > 1} onto Ω.

Proof. Our main idea is an approximation of the function log f′ by lacunary series. Let

ψ(ζ) = log \frac{F′(f(ζ))}{qζ^{q−1}} = ∑_{k=1}^{q−1} log \frac{f(ζ) − ζ_{k}}{ζ} = ∑_{j=0}^{∞} b_{j}ζ^{−j}.

It is known [7] that

log f′(ζ) = − ∑_{k=0}^{∞} ψ(ζ^{q^{k}}) = φ(ζ) + g(ζ),

where g(ζ) = ∑_{j=N+1}^{∞} ∑_{k=0}^{∞} b_{j}ζ^{−jq^{k}} and φ(ζ) = ∑_{j=0}^{N} ∑_{k=0}^{∞} b_{j}ζ^{−jq^{k}}. Fixing ε > 0, we will show that there exists N = N(ε) such that |ζg′(ζ)| ≤ ε/(|ζ|^{2} − 1). From Pommerenke’s result [8, p. 100] it follows that ∑_{j=0}^{∞} |b_{jk}|^{2}j^{1+α} < +∞ for John domains, where b_{jk} are the Taylor coefficients of log(f(ζ) − ζ_{k})/ζ. This implies that ∑_{j=0}^{∞} |b_{j}|^{2}j^{1+α} < +∞ for John domains. Therefore, we have

|ζg′(ζ)| = | ∑_{k=0}^{∞} ∑_{j=N+1}^{∞} jq^{k}b_{j}ζ^{−jq^{k}}| ≤ ∑_{k=0}^{∞} ∑_{j=N+1}^{∞} jq^{k}|b_{j}|R^{−jq^{k}}

≤ ∑_{k=0}^{∞} q^{k}\sqrt{∑_{j=N+1}^{∞} j^{1+α}|b_{j}|^{2}} \sqrt{∑_{j=N+1}^{∞} j^{1−α}R^{−2jq^{k}}}

≤ ε_{N,α} ∑_{k=0}^{∞} \frac{q^{k}R^{−q^{k}}}{(1 − R^{−q^{k}})^{α/2}} = ε_{N,α} ∑_{k=0}^{∞} \frac{q^{k}r^{q^{k}}}{(1 − r^{q^{k}})^{α/2}}

≤ ε_{N,α} ∑_{k=0}^{∞} \frac{q^{k}r^{q^{k}}}{(1 − r)^{α/2}q^{αk/2}r^{αq^{k}/2}}

= \frac{ε_{N,α}}{(1 − r)^{α/2}} ∑_{k=0}^{∞} q^{(1−α/2)k}r^{(1−α/2)q^{k}} ≤ C \frac{ε_{N,α}}{1 − r} , where r = \frac{1}{R} < 1.

So, there exists N = N(α, ε) such that |ζg′(ζ)| ≤ ε/(|ζ|^{2} − 1). Consider now

∫ |f′(Re^{iθ})|^{t} dθ = ∫ |e^{φ}|^{t}|e^{g}|^{t} dθ ≤ (∫ |e^{φ}|^{pt} dθ)^{1/p}(∫ |e^{g}|^{st} dθ)^{1/s},

where s = 1/ε, p = 1/(1 − ε).

Since |ζg′(ζ)| ≤ ε/(|ζ|^{2} − 1) then it follows from the result of Clunie and Pommerenke [3] that

∫ |e^{g}|^{st} dθ ≤ C(\frac{1}{R − 1})^{t^{2}/4}.

It is easy to see that φ is a lacunary series with Hadamard gaps. Applying Theorem 2 to this series we see that

∫ |e^{φ}|^{pt} dθ ≍ (\frac{1}{R − 1})^{t^{2}p^{2}σ_{φ}^{2}+O_{N} (t^{2+1/5})},

where

σ_{φ}^{2} = \frac{1}{2π} lim sup_{R→1} \frac{∫ |φ|^{2} dθ}{log(1/(R − 1))}

Thus,

∫ |f′(Re^{iθ})|^{t} dθ ≤ C(\frac{1}{R − 1})^{t^{2}(σ_{φ}^{2}+O(ε))+O_{N} (t^{2+1/5})}.

Arguing as above, we obtain

∫ |f′(Re^{iθ})|^{t} dθ ≥ C(\frac{1}{R − 1})^{t^{2}(σ_{φ}^{2}+O(ε))+O_{N} (t^{2+1/5})}.

Obviously, σ_{φ} → σ as ε → 0. Hence, σ(0+)^{2} = σ^{2} .

364 I. R. Kayumov

Corollary. Let Ω be a simply connected John domain. Then

\overline{lim_{R→1}} \frac{| log f′(Rζ)|}{\sqrt{log(1/(R − 1)) log log log(1/(R − 1))}} = \sqrt{σ^{2}(0+)}

for almost all ζ on |ζ| = 1.

Proof. This formula immediately follows from Theorem 3 and the well-known law of the iterated logarithm for Julia sets [9]:

\overline{lim_{R→1}} \frac{| log f′(Rζ)|}{\sqrt{log(1/(R − 1)) log log log(1/(R − 1))}} = \sqrt{σ^{2}} .

Note also that this equality follows from the classical law of the iterated logarithm for lacunary series [10]. Therefore, we see again that (2) is sharp.

Acknowledgements. I want to thank Professors Christian Pommerenke, Stephan Ruscheweyh and Richard Furnier for their useful remarks, and Professor Farit Avkhadiev for helpful discussions.

This work was supported by the German Academic Exchange Service (DAAD, University of Würzburg) and by Russian Fund of Basic Research (Grants N 99-01-00366, 99-01-00173).

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[10] Weiss, M.: On the law of the iterated logarithm for lacunary trigonometric series. - Trans. Amer. Math. Soc. 91, 1959, 444–469.

Received 10 September 2001