Löwner expansions (Q801434)

Let S denote the class of functions \(f(z)=z+a_ 2z^ 2+..\). that are analytic and univalent in the unit disk. The author recently proved [Acta Math. 154, 137-152 (1985)] the famous Bieberbach conjecture that \(| a_ n| \leq n\) for all n via proving the Lebedev-Milin conjecture on sums of squares of the coefficients of \(\log [f(z)/z].\) The paper under review is a forerunner proving weaker results. It is quite interesting to read for this reason. It contains already most of the ideas of the later proof except for one major ingredient, namely the de Branges system of special functions. The paper shows how de Branges' ideas grew out of his work on square summable power series using Löwner's method of composing functions in the unit disk.