the law of the iterated logarithm for locally univalent functions

Annales Academiæ Scientiarum FennicæMathematicaVolumen 27, 2002, 357–364THE LAW OF THE ITERATED LOGARITHMFOR LOCALLY UNIVALENT FUNCTIONSI. R. KayumovKazan State University, Chebotarev Institute of Mathematics and MechanicsUniversitetskaya 17, Kazan 420008, Russia; ikayumov@ksu.ruAbstract. 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 anasymptotic behaviour of the integral means of the derivative of an analytic function. Also, weestablish that this constant is equal to the asymptotic variance for some domains with fractal typeboundaries.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 supr→1−| log f ′ (rζ)|√log ( 1/(1 − r) ) log log log ( 1/(1 − r) ) ≤ C‖ log f ′ ‖ Bfor almost all ζ on |ζ| = 1, where‖ log f ′ ‖ B = | log f ′ (0)| + sup (1 − |z| 2 )|z|

358 I. R. KayumovThe goal of this paper is to obtain a sharp version of the Makarov law of theiterated logarithm for locally univalent functions, i.e. for functions f for whichf ′ (z) ≠ 0, z ∈ D .Let f be a locally univalent function in the unit disk D and p be a complexnumber. Then for all δ > 0 we defineβ δ (p) =logsupr∈[0,1)[δ ∫ ]|z|=r |f ′ (z) p |dθlog ( 1/(1 − r) ) .In other words β δ (p) is the minimal number for whichIf p is a real number thenwhere∫|f ′p | dθ ≤ 1 δ( ) βδ (p) 1, 0 ≤ r < 1.1 − rβ δ (p) → β(p) as δ → 0,β(p) = lim supr→1log ∫ |z|=r |f ′ (z)| p dθlog ( 1/(1 − r) )is the classical integral means spectrum [8, p. 176].It follows from the integral means spectrum concept ( [2], [7], [8]) that thereis a connection between geometric properties of domains and the integral meansspectrum. On the other hand, Makarov [5] established that the law of the iteratedlogarithm is closely related to the boundary properties of conformal maps. Thisleads to the following natural question: Is there a simple relation between the lawof the iterated logarithm and the integral means spectrum? A possible answer forthis question is the following result which we will prove later.Suppose f is a locally univalent function in the unit disk and δ > 0. Thenlim supr→1−| log f ′ (rζ)|√log ( 1/(1 − r) ) log log log ( ) ≤ 2 lim sup1/(1 − r)p→0√βδ (p)for almost all ζ on |ζ| = 1.It is more convenient for us to formulate this result in the form of the followingTheorem 1. Suppose f is a locally univalent function in the unit disk. Thenthe following inequality holds(2) lim supr→1−| log f ′ (rζ)|√log ( 1/(1 − r) ) log log log ( 1/(1 − r) ) ≤ √ σ 2 (0+)|p|

The law of the iterated logarithm for locally univalent functions 359for almost all ζ on |ζ| = 1, whereσ 2 (δ) = 4 lim supp→0β δ (p)|p| 2 .We remark that Pommerenke’s result with the constant C = 1 easily followsfrom our theorem because it is known [3] that if log f ′ is a Bloch function thenσ 2 (0+) ≤ ‖ log f ′ ‖ 2 B . Moreover, we will see that in many casesσ 2 (0+) = σ 2 .Further, it is convenient to use the following abbreviation:∫ 2π∫h(re iθ ) dθ ≡ h dθ0The next lemma can be deduced from Makarov’s proof of the law of theiterated logarithm [5].Lemma 1. Let C k be a sequence of positive numbers and C 1/kk→ 1 ask → ∞. If∫| log f ′ | 2n dθ ≤ C n n!A 2n log n 11 − rfor all natural n and for all r ∈ [ 1 − exp(− exp e n ), 1 ) , thenlim supr→1−for almost all ζ on |ζ| = 1.| log f ′ (rζ)|√log ( 1/(1 − r) ) log log log ( 1/(1 − r) ) ≤ AProof. 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∞in his proof we have to consider ∫ ∞eexp(e n ) .Proof of Theorem 1. Fix δ > 0 and ε > 0. Then there exists p 0 = p 0 (δ, ε) > 0such that∫|f ′p | dθ ≤ 1 ( ) (σ 12 (δ)+ε)|p| 2 /4δ 1 − rfor |p| < p 0 . This implies∫∫I 0 (t| log f ′ |) =≤ 1 δ∫12πt( 11 − r|f ′p | |dp| dθ = 1 ∫ ∫2πt |p|=t) (σ 2 (δ)+ε)t 2 /4, t ∈ (0, p 0 ),|p|=t|f ′p | d θ|dp|

360 I. R. KayumovwhereI 0 (x) =∞∑( x2k=04) k/k! 2 = 1 ∫ 2πe x cos θ dθ2π 0is the modified Bessel function of order zero [1]. At the same time,∫Setting| log f ′ | 2n dθ ≤ 4nt 2n n!2 ∫t 2 =and using the identitywe obtain∫Applying Lemma 1, we getlim supr→1−I 0 (t| log f ′ |) dθ ≤ 4n 1 ( ) (σ 12 (δ)+ε)t 2 /4t 2n n!2 .δ 1 − r4n(σ 2 (δ) + ε) log ( 1/(1 − r) ), n ≤ log log log 11 − r ,e =( 1) 1/ log(1/(1−r))1 − r| log f ′ | 2n dθ ≤ 1 δ n!2 e n 1 (σ 2n n (δ) + ε ) ( ) nn 1log .1 − r| log f ′ (rζ)|√log ( 1/(1 − r) ) log log log ( 1/(1 − r) ) ≤ √ σ 2 (δ) + εfor almost all ζ on |ζ| = 1 and for all δ > 0, ε > 0. Hence, this result is true forδ = 0+ and ε = 0. This completes the proof.If f is a univalent function and f(D) is a domain with rectifiable boundarythen inequality (2) is trivially sharp. Non-trivial examples, which show that thisinequality is sharp, can be obtained by using lacunary series.Let log f ′ = ∑ ∞k=1 a kz n kbe a lacunary series with bounded coefficients andn k+1 /n k ≥ q > 1. Since log f ′ is a Bloch function then σ 2 (0+) < +∞ as wasmentioned above. In the other direction, Makarov [6] showed that if n k = 2 k anda k = 1 for all k then σ 2 (0+) > 0. Rohde [8] improved his result in the followingsense. 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 thenσ 2 (0+) = lim supr→1B 2log ( 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 thefollowing

The law of the iterated logarithm for locally univalent functions 361Definition ([10]). We say that a lacunary series satisfies condition (ϱ, R) ifit consists of blocks of terms of length R, separated by empty blocks of length ϱ.Weiss [10] proved the followingLemma 2. Let f(z) = ∑ ∞k=1 a kz n kbe a lacunary series satisfying condition(ϱ, R), where(q ϱ/3 − 1) −1 ≤ 1 4 (q − 1) and q−R/3 (R + 1) 2 ≤ 1 2 .Let M = sup |a k |, B 2 = ∑ ∞n=1 |a k| 2 r 2n k, r ∈ [0, 1). Then∫πe (1−ctR2 )t 2 B 2 /4 ≤ e t Re f(reiθ) dθ ≤ 3πe (1+ctR2 )t 2 B 2 /4 .Denote by ϱ 0 the minimal positive number for which(q ϱ 0/3 − 1) −1 ≤ 1 4 (q − 1) and q−ϱ 0/3 (ϱ 0 + 1) 2 ≤ 1 2 .Now, we can prove the followingTheorem 2. Let log f ′ = ∑ ∞k=1 a kz n kbe a lacunary series with boundedcoefficiens for which n k+1 /n k ≥ q > 1. Thenσ 2 (0+) = lim supr→1B 2log ( 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 satisfiesthe (ϱ 0 , R)-condition and f 2 satisfies the (R, ϱ 0 )-condition. Let us estimate B12and B2 2 . In fact, it is enough to estimate only B 2 because B 2 = B1 2 + B2 2 . WehaveB 2 ≤ M 2∞∑j=1j(R+ϱ 0 )∑∑ ∞r 2qk ≤ M 2 ϱ 0k=jR+(j−1)ϱ 0 j=1r 2qRj≤ C log11 − r /R.Evidently, without loss of generality, we can assume that p = t is a real positivenumber. Setting R = t −2/5 , we have∫ ∫(∫) 1/α (∫1/β|f ′ | t dθ = e t Re f 1+t Re f 2dθ ≤ e αt Re f 1dθ e βt Re f 2dθ),where β = t −1/5 and α = (1 − t 1/5 ) −1 . Applying Lemma 2 with f 1 and f 2 andthe estimate for B 2 we obtain∫( ) Ct2+1/5|f ′ | t dθ ≤ Ce B2 t 2 /4 1.1 − rAnalogously,This concludes the proof.∫|f ′ | t dθ ≥ Ce B2 t 2 /4 (1 − r) Ct2+1/5 .

362 I. R. KayumovApplying the law of the iterated logarithm for lacunary series [10] it is easyto show that ifB 2limr→1 log ( 1/(1 − r) )exists then the equality in (2) holds. Other examples which show that (2) is sharpcome from the theory of Julia sets. The idea for using Julia sets in the theory ofunivalent functions is due to Carleson and Jones [2]. They conjectured that thebasin 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 andbe the basin of attraction of ∞ for F .Ω = {ζ : F on (ζ) → ∞ as n → ∞}Theorem 3. Let Ω be a simply connected John domain. Thenσ 2 (0+) = σ 2 = lim supr→1B 2log ( 1/(1 − r) ),where B 2 = ∑ |a k | 2 r k ; a k are the coefficients of log f ′ and f is the conformalmapping from D − = {|ζ| > 1} onto Ω.Proof. Our main idea is an approximation of the function log f ′ by lacunaryseries. Letψ(ζ) = log F ′( f(ζ) ) ∑q−1qζ q−1 =k=1log f(ζ) − ζ kζ∞∑= b j ζ −j .j=0It is known [7] thatlog f ′ (ζ) = −∞∑ψ(ζ qk ) = ϕ(ζ) + g(ζ),k=0where g(ζ) = ∑ ∞ ∑ ∞j=N+1 k=0 b jζ −jqk and ϕ(ζ) = ∑ N ∑ ∞j=0 k=0 b jζ −jqk . Fixingε > 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 < +∞ forJohn domains, where b jk are the Taylor coefficients of log(f(ζ) − ζ k )/ζ . This

The law of the iterated logarithm for locally univalent functions 363implies that ∑ ∞j=0 |b j| 2 j 1+α < +∞ for John domains. Therefore, we have∞∑ ∞∑∞∑ ∞∑|ζg ′ (ζ)| = | jq k b j ζ −jqk | ≤ jq k |b j |R −jqk≤k=0 j=N+1√∞∑ √√√q kk=0≤ ε N,α≤ ε N,α=∞∑k=0∑∞∞∑j=N+1k=0ε N,α∑ ∞(1 − r) α/2j 1+α |b j | 2 √ √√√k=0 j=N+1∞∑j=N+1q k R −qk(1 − R −qk ) α/2 = ε N,αq k r qk(1 − r) α/2 q αk/2 r αqk /2k=0∞∑k=0q (1−α/2)k r (1−α/2)qkj 1−α R −2jqkq k r qk(1 − r qk ) α/2≤ C ε N,α1 − r , where r = 1 R < 1.So, there exists N = N(α, ε) such that |ζg ′ (ζ)| ≤ ε/(|ζ| 2 − 1). Consider now∫∫(∫ ) 1/p (∫1/s|f ′ (Re iθ )| t dθ = |e ϕ | t |e g | t dθ ≤ |e ϕ | pt dθ |e g | dθ) st ,where s = 1/ε, p = 1/(1 − ε).Since |ζg ′ (ζ)| ≤ ε/(|ζ| 2 − 1) then it follows from the result of Clunie andPommerenke [3] that∫( ) t 12 /4|e g | st dθ ≤ C.R − 1It is easy to see that ϕ is a lacunary series with Hadamard gaps. ApplyingTheorem 2 to this series we see that∫|e ϕ | pt dθ ≍where( 1R − 1) t 2 p 2 σ 2 ϕ +O N (t 2+1/5 ),σϕ 2 = 1∫|ϕ| 2 dθlim sup2π R→1 log ( 1/(R − 1) )Thus,∫( ) t 12 (σ 2|f ′ (Re iθ )| t ϕ +O(ε))+O N (t 2+1/5 )dθ ≤ C.R − 1Arguing as above, we obtain∫( ) t 12 (σ 2 ϕ +O(ε))+O N (t 2+1/5 )|f ′ (Re iθ )| t dθ ≥ C.R − 1Obviously, σ ϕ → σ as ε → 0. Hence, σ(0+) 2 = σ 2 .

364 I. R. KayumovCorollary. Let Ω be a simply connected John domain. Then| log f ′ (Rζ)|lim √log ( 1/(R − 1) ) log log log ( 1/(R − 1) ) = √ σ 2 (0+)R→1for almost all ζ on |ζ| = 1.Proof. This formula immediately follows from Theorem 3 and the well-knownlaw of the iterated logarithm for Julia sets [9]:| log f ′ (Rζ)|lim √log ( 1/(R − 1) ) log log log ( 1/(R − 1) ) = √ σ 2 .R→1Note also that this equality follows from the classical law of the iterated logarithmfor lacunary series [10]. Therefore, we see again that (2) is sharp.Acknowledgements. I want to thank Professors Christian Pommerenke, StephanRuscheweyh and Richard Furnier for their useful remarks, and Professor FaritAvkhadiev 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).References[1] Abramowitz, M., and I. Stegun: Handbook of Mathematical Functions. - Dover, NewYork, 1972.[2] Carleson, L., and P.W. Jones: On coefficient problems for univalent functions andconformal dimension. - Duke Math. J. 66, 1992, 169–206.[3] Clunie, J., and Ch. Pommerenke: On the coefficients of univalent functions. - MichiganMath. J. 14, 1967, 71–78.[4] Kayumov, I.R.: The integral means spectrum for lacunary series. - Ann. Acad. Sci. Fenn.Math. 26, 2001, 447–453.[5] Makarov, N.G.: On the distortion of boundary sets under conformal mappings. - Proc.London Math. Soc. 51, 1985, 369–384.[6] Makarov, N.G.: A note on integral means of the derivative in conformal mapping. -Proc. Amer. Math. Soc. 96, 1986, 233–235.[7] Makarov, N.G.: Fine structure of harmonic measure. - St. Petersburg Math. J. 10, 1999,217–268.[8] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. - Springer-Verlag, Berlin,1992.[9] Przytycki, F., M. Urbanski, and A. Zdunik: Harmonic, Gibbs, and Hausdorff measureson repellers for holomorphic maps. I. - Ann. of Math. 2, 1989, 1–40; II, StudiaMath. 97, 1991, 189–225.[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