On the Bombieri numbers for the class S
Let be \[ f(z)=z+a_2 z^2+a_3 z^3 +\cdots + a_n z^n +\cdots \] analytic and univalent in the unit disk, then the Bombieri numbers are defined in the following way \[ \sigma_{mn} := \liminf_{a_m \to m} \frac{n - Re ( a_n )}{m - Re ( a_m )}. \] The question posed by \textit{E. Bombieri} [Invent. Math. 4, 26--67 (1967; Zbl 0174.12301)] is: are the Bombieri numbers equal to the numbers \[ B_{mn} := \min_{\theta} \frac{n \sin \theta -\sin {n\theta}}{m\sin \theta -\sin {m\theta}}. \] The aim of the paper is to show that this is wrong in many cases, more precisely it is proven that \(\sigma_{m2}< B_{m2} \) for all \(m>2\) and \(\sigma_{m3} <B_{m3} \) for all odd \(m>3\). Furthermore a new derivation of the second variation formula of Bombieri on the coefficient \(a_n\) near the Koebe function is given.
scientific article; zbMATH DE number 6711589
scientific article