the law of the iterated logarithm for locally univalent functions

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