On coefficient problems for univalent functions and conformal dimension (Q1196384)

Let \(\mathbb{D}\) and \(\Delta\) denote the unit disk and its exterior, and let \[ S_ 1:=\left\{f(z)=\sum_{n=0}^ \infty a_ n z^ n:\;\mathbb{D}\to\mathbb{C}\mid\;f \text{ is univalent and }| f(z)|<1\right\} \] and \[ S_ 2:=\left\{f(z)=z+\sum_{n=2} ^ \infty a_ n z^{-n}: \Delta\to\mathbb{C}\mid\;f\text{ is univalent}\right\} \] denote the families of bounded univalent, and meromorphic univalent functions, respectively. The coefficient problems to determine \[ \gamma_ 1:=- \lim_{n\to\infty} {{\log \sup_{f\in S_ 1} | a_ n|} \over {\log n}} \qquad \text{and} \qquad \gamma_ 2:=-\lim_{n\to\infty} {{\log \sup_{f\in S_ 2} | a_ n|} \over {\log n}} \] both are unsolved and are connected with each other. The authors show that \(\gamma_ 1=\gamma_ 2\), and conjecture that they equal \(3/4\). In order to obtain good estimates for \(\gamma_ 1\) and \(\gamma_ 2\) they construct mappings with ``fractal boundary'' and show that the given limits can be approached by those functions.