Treffer: The integral means spectrum for lacunary series

Let \(f\) be a univalent function in the unit disk \(D\), and let \(\log{f'(z)} = \sum_{k=1}^\infty a_k z^{n_k}\) be a lacunary series with bounded coefficients and \(n_{k+1}/n_k \geq \lambda \geq 2\). The integral means spectrum of \(f\) is defined by \[ \beta_f(t) := \varlimsup_{r \to 1} {\frac{\log{\int_{|z|=r}|f'(re^{i\theta})|^td\theta}} {\log{\frac{1}{1-r}}}} , \quad t>0 . \] The main result of this paper is the asymptotic formula \[ \beta_f(t) = \frac{t^2}{4} \varlimsup_{r \to 1} {\frac{b^2(r)}{\log{\frac{1}{1-r}}}} + O(t^{5/2}) \quad \text{as} \quad t \to 0 , \] where \[ b^2(r) := \sum_{k=1}^\infty |a_k|r^{2n_k}. \] From this and the law of the iterated logarithm the author deduces the estimate \[ \varlimsup_{r \to 1}{\frac{|\log{f'(r\zeta)}|} {\sqrt{\log{\frac{1}{1-r}}\log{\log{\log{\frac{1}{1-r}}}}}}} \leq 2 \lim_{t \to 0}{\frac{\sqrt{\beta(t)}}{t}} \] for almost all \(\zeta\) with \(|\zeta|=1\). Equality holds if there exists \[ \lim_{r \to 1}{\frac{b^2(r)}{\log{\frac{1}{1-r}}}} . \]