Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 447–453

THE INTEGRAL MEANS SPECTRUM FOR LACUNARY SERIES

I. R. Kayumov

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

Abstract. Asymptotically sharp bounds for the integral means spectrum of lacunary series are proved. In particular, we show that Rohde’s estimates for lacunary series with positive coefficients are sharp and hold not only for the positive case. Moreover, a relation between the law of the iterated logarithm and the integral means spectrum is established. Using this we give a sharp version of the Makarov law of the iterated logarithm for lacunary series.

1. Introduction

Let log f′(z) = ∑_{k=1}^{∞} a_{k}z^{n_{k}} be a lacunary series with bounded coefficients and n_{k+1}/n_{k} ≥ λ ≥ 2 and

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

be the integral means spectrum.

In this paper we show that

\overline{lim_{r→1}} \frac{| log f′(rζ)|}{\sqrt{log \frac{1}{1 − r} log log log \frac{1}{1 − r}}} ≤ 2 lim_{t→0} \frac{\sqrt{β(t)}}{t}

for almost all ζ on |ζ| = 1. The equality holds if there exists

lim_{r→1} \frac{b^{2}(r)}{log \frac{1}{1 − r}} where b^{2}(r) = ∑_{k=1}^{∞} |a_{k}|^{2}r^{2n_{k}}.

This result follows from the law of the iterated logarithm and the asymptotic formula

β(t) = \frac{t^{2}}{4} \overline{lim_{r→1}} \frac{b^{2}(r)}{log \frac{1}{1 − r}} + O(t^{2.5}) as t → 0

1991 Mathematics Subject Classification: Primary 30C55.

which will be proved later. The law of the iterated logarithm, proved by Weiss [11], states that

\overline{lim_{r→1}} \frac{| log f′(rζ)|}{b(r)\sqrt{log log b(r)}} = 1 for almost all |ζ| = 1.

A more general result was obtained by Makarov [5]:

\overline{lim_{r→1}} \frac{| log f′(rζ)|}{\sqrt{log \frac{1}{1 − r} log log log \frac{1}{1 − r}}} ≤ ∥ log f′∥_{B}

for almost all ζ on |ζ| = 1 and all functions log f′ ∈ B. Here B is the class of Bloch functions, i.e., analytic in the unit disk D = {|z| < 1} functions g with the norm

∥g∥_{B} = sup_{z∈D} |g′(z)|(1 − |z|^{2}) < +∞.

One of the main problems of the boundary behaviour of conformal maps is the investigation of the integral means spectrum β_{f}(t) for univalent functions f ([2], [3], [4], [7] and [9]). Since the properties of Bloch functions are similar to the properties of lacunary series it is helpful to study β_{f}(t) for lacunary series because it is well known that, for any conformal mapping f, log f′ is a Bloch function.

The first non-trivial result about the integral means spectrum for lacunary series was obtained by Makarov [6]. He showed that if f is defined by

log f′(z) = \frac{i}{5} ∑_{k=1}^{∞} z^{2^{k}}

then β_{f}(t) ≥ 0.00035t^{2} for small t. Using Bessel functions Rohde [10], [8] improved this result. He obtained that β_{f}(t) ≥ log I_{0}(at)/ log q for the lacunary series

log f′(z) = a ∑_{k=1}^{∞} z^{q^{k}}, a > 0

where I_{0}(x) is the modified Bessel function.

First we prove some auxiliary results. It is convenient to use the following abbreviation

∫ h dθ ≡ ∫_{0}^{2π} h(re^{iθ}) dθ.

Lemma 1. Let log f′, log φ′ ∈ B. Then there exists C > 0 such that

\frac{1}{C} (1 − r)^{t^{2.5}} ∫ |f′|^{t} dθ ≤ ∫ |g′| dθ ≤ C(\frac{1}{1 − r} )^{Ct^{2.5}} ∫ |f′|^{t} dθ,

where f′^{t} = g′φ′^{t^{2}} and t < 1.

The integral means spectrum for lacunary series 449

Proof. It is known [8], [1] that there exists C > 0 such that

|f′|^{t}, |g′| ≤ C(\frac{1}{1 − r} )^{Ct},

∫ |f′|^{t} dθ, ∫ |φ′|^{±t} dθ, ∫ |g′| dθ ≤ C(\frac{1}{1 − r} )^{Ct^{2}}.

Using the Hölder inequality we have

∫ |f′|^{t} dθ = ∫ |g′| |φ′|^{t^{2}} dθ

≤ (∫ |g′|^{1/(1−t^{3/2})} dθ)^{1−t^{3/2}}(∫ |φ′|^{t^{1/2}} dθ)^{t^{3/2}}

≤ (sup |g′|)^{t^{3/2}}(∫ |g′| dθ)^{1−t^{3/2}}C(\frac{1}{1 − r} )^{Ct^{5/2}}

≤ C^{2}(\frac{1}{1 − r} )^{2Ct^{5/2}} ∫ |g′| dθ(∫ |g′| dθ)^{t^{3/2}}

≤ C^{3}(\frac{1}{1 − r} )^{3Ct^{5/2}} ∫ |g′| dθ.

Analogously applying the Hölder inequality to ∫ |f′|^{t}|φ′|^{−t^{2}} dθ we obtain

∫ |g′| dθ = ∫ |f′|^{t}|φ′|^{−t^{2}} dθ ≤ C^{3}(\frac{1}{1 − r} )^{3Ct^{5/2}} ∫ |f′|^{t} dθ.

Lemma 1 is proved.

Lemma 2. Let log f′ = ∑_{k=1}^{∞} a_{k}z^{k} ∈ B. Then there exists C > 0 such that

\frac{1}{C} (1 − r)^{Ct^{5/2}}I(r, t) ≤ ∫ |f′|^{t} dθ ≤ C(\frac{1}{1 − r} )^{Ct^{5/2}}I(r, t)

where

I(r, t) = ∫ ∏_{k=1}^{∞} |1 + a_{k}tz^{k}/2|^{2} dθ.

Proof. We have

t log f′ = ∑_{k=1}^{∞} ta_{k}z^{k} = ∑_{k=1}^{∞} 2 log(1 + ta_{k}z^{k}/2) + ∑_{k=1}^{∞} t^{2}a_{k}^{2}z^{2k}/4 + ∑_{k=1}^{∞} O((ta_{k}r^{k})^{3}).

From [8], [1] it follows that log φ′ = ∑_{k=1}^{∞} a_{k}^{2}z^{2k} ∈ B and ∑_{k=1}^{∞} O((a_{k}r^{k})^{3}) = O (log(1/(1 − r))). Therefore, we can apply Lemma 1 and obtain the required inequalities.

450 I. R. Kayumov

Now we can prove our main result.

Theorem 1. Let log f′ = ∑_{k=1}^{∞} a_{k}z^{n_{k}} be a lacunary series with bounded coefficients and n_{k+1}/n_{k} ≥ λ ≥ 2.

Then

β_{f}(t) = \frac{t^{2}}{4} \overline{lim_{r→1}} \frac{∑_{k=1}^{∞} |a_{k}|^{2}r^{2n_{k}}}{log \frac{1}{1 − r}} + O(t^{5/2}) as t → 0.

Proof. By Lemma 2, to prove this result it is enough to estimate

∫ ∏_{k=1}^{∞} |1 + ta_{k}z^{n_{k}}/2|^{2} dθ = ∫ ∏_{k=1}^{∞}(1 + t^{2}|a_{k}|^{2}r^{2n_{k}}/4 + t|a_{k}|r^{n_{k}} cos(n_{k}θ + θ_{k})) dθ.

We multiply out and integrate term-by-term. Since

cos α cos β = 0.5[cos(α + β) + cos(α − β)]

then

(1)

cos(n_{p_{1}}θ + θ_{p_{1}}) cos(n_{p_{2}}θ + θ_{p_{2}}) × ⋯ × cos(n_{p_{j}}θ + θ_{p_{j}})

= \frac{1}{2^{j}} ∑ cos((n_{p_{1}} ± n_{p_{2}} ± ⋯ ± n_{p_{j}})θ + γ).

The integral of (1) by θ is equal to zero because ∑_{k=1}^{j−1} n_{p_{k}} ≤ ∑_{k=1}^{p_{j}−1} n_{k} < n_{p_{j}}/(λ − 1) ≤ n_{p_{j}}. Therefore

∫ ∏_{k=1}^{∞} |1 + a_{k}tz^{n_{k}}/2|^{2} dθ = 2π ∏_{k=1}^{∞}(1 + t^{2}|a_{k}|^{2}r^{2n_{k}}/4)

= 2π exp[∑_{k=1}^{∞} log(1 + t^{2}|a_{k}|^{2}r^{2n_{k}}/4)]

= 2π(\frac{1}{1 − r} )^{O(t^{3})} exp[0.25t^{2} ∑_{k=1}^{∞} |a_{k}|^{2}r^{2n_{k}}].

This concludes the proof.

Applying Theorem 1 to the law of the iterated logarithm we obtain Theorem 2. For almost all ζ on |ζ| = 1 the following inequality holds

\overline{lim_{r→1}} \frac{| log f′(rζ)|}{\sqrt{log \frac{1}{1 − r} log log log \frac{1}{1 − r}}} ≤ 2 lim_{t→0} \frac{\sqrt{β(t)}}{t}.

The inequality is sharp, i.e., if lim_{r→1}(b^{2}(r)/log(1/(1 − r))) exists then equality holds.

Now we will prove an estimate for a partial case n_{k} = q^{k}, q is an integer. To prove this result we need some lemmas.

Lemma 3. Let

I_{n}(x) = (\frac{x}{2})^{n} ∑_{ν=0}^{∞} \frac{x^{2ν}}{4^{ν}ν!(ν + n)!} (n = 0, 1, 2, …)

be modified Bessel functions. Then

x cos θ − log(I_{0}(x) + 2 ∑_{n=1}^{q−1} I_{n}(x) cos(nθ)) = O(x^{q}) as x → 0.

Proof. It is known [8] that

exp(x cos θ) = I_{0}(x) + 2 ∑_{n=1}^{∞} I_{n}(x) cos(nθ).

Hence

exp(x cos θ) = I_{0}(x) + 2 ∑_{n=1}^{q−1} I_{n}(x) cos(nθ) + O(x^{q})

because I_{n}(x) ≤ (|x|^{n}/2^{n}) exp(|x|^{2}/4)/n!. Therefore

x cos θ = log(I_{0}(x) + 2 ∑_{n=1}^{q−1} I_{n}(x) cos(nθ))

+ log(1 + \frac{O(x^{q})}{I_{0}(x) + 2 ∑_{n=1}^{q−1} I_{n}(x) cos(nθ)})

= log(I_{0}(x) + 2 ∑_{n=1}^{q−1} I_{n}(x) cos(nθ)) + O(x^{q}).

Lemma 4. Let q > 1, s_{ij} ≠ 0, p_{ij} > 0 be some integers and |s_{ij}| < q, p_{ij} ≠ p_{ls}, (i, j) ≠ (l, s). Then

(2)  ∫ ∏_{i=1}^{m} cos(θ ∑_{j=1}^{n_{i}} s_{ij}q^{p_{ij}} + θ_{i}) dθ = 0.

Proof. The proof is by induction on m. Consider the case m = 1. Suppose that

∫ cos(θ ∑_{j=1}^{n} s_{j}q^{p_{j}} + θ_{1}) dθ ≠ 0.

Then ∑_{j=1}^{n} s_{j}q^{p_{j}} = 0. Without loss of generality we can suppose p_{1} < p_{2} ⋯ < p_{n}. Multiplying our equation on q^{−(p_{1}+1)} we conclude that s_{1}/q is an integer number but this is impossible because 0 < |s_{1}| < q.

Now suppose that m ≥ 2 and (2) holds for m − 1. Using cos α cos β = 0.5[cos(α + β) + cos(α − β)] we can write our integral as the sum of two integrals with m − 1 factors and hence our hypothesis is true.

Theorem 3. We have

β_{f}(t) = \overline{lim_{r→1}} \frac{∑_{k=1}^{∞} log I_{0}(t|a_{k}|r^{q^{k}})}{log \frac{1}{1 − r}} + O(t^{q}) as t → 0.

Proof. Since

∫ |f′(z)|^{t} dθ = ∫ exp[t ∑_{k=1}^{∞} |a_{k}|r^{q^{k}} cos(q^{k}θ + θ_{k})] dθ

= ∫ ∏_{k=1}^{∞} exp[t|a_{k}|r^{q^{k}} cos(q^{k}θ + θ_{k})] dθ.

it follows from Lemma 3 that

∫ ∏_{k=1}^{∞} exp[t|a_{k}|r^{q^{k}} cos(q^{k}θ + θ_{k})] dθ

= (\frac{1}{1 − r} )^{O(t^{q})} ∫ ∏_{k=1}^{∞}(I_{0}(t|a_{k}|r^{q^{k}}) + 2 ∑_{j=1}^{q−1} I_{j}(t|a_{k}|r^{q^{k}}) cos j(q^{k}θ + θ_{k})) dθ

where we have used that ∑_{k=1}^{∞} r^{q^{k}} = O(log 1/(1 − r)).

We multiply out and integrate term-by-term. By Lemma 4 our integral is

2π ∏_{k=1}^{∞} I_{0}(t|a_{k}|r^{q^{k}}).

Theorem 3 is proved.

Corollary. Let |a_{k}| = a > 0 then

β_{f}(t) = \frac{log I_{0}(at)}{log q} + O(t^{q}) as t → 0.

Proof. From the proof of Rohde it follows easily that

\overline{lim_{r→1}} \frac{∑_{k=1}^{∞} log I_{0}(tar^{q^{k}})}{log \frac{1}{1 − r}} = \frac{log I_{0}(at)}{log q} + O(t^{q}) as t → 0.

The integral means spectrum for lacunary series 453

Applying Theorem 3 we conclude that

β_{f}(t) = \overline{lim_{r→1}} \frac{∑_{k=1}^{∞} log I_{0}(atr^{q^{k}})}{log \frac{1}{1 − r}} + O(t^{q}) = \frac{log I_{0}(at)}{log q} + O(t^{q}) as t → 0.

Acknowledgements. I thank Professors F.G. Avhadiev and S.R. Nasyrov for helpful discussions.

This work was supported by Russian Fund of Fundamental Research (proj. 99-01-00366, 99-01-00173).

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Received 2 March 2000