ljdickey@watmath.UUCP (Lee Dickey) (08/24/84)
A recent "Informal Communication" from Carl H. Fitzgerald (San Diego) and Ch. Pommerenke (Berlin) gives a short (nine page) proof of ``The de Branges Theorem on Univalent Functions'', which settles the long- standing (since 1916) conjecture of Bieberbach about analytic and univalent functions in the unit disk. Their paper represents a distillation of notes prepared by I.M.Milin (Leningrad) of lectures presented by de Branges (Purdue). The problem is easily stated: a function of the form f(z) = z + sum from 2 to infinity a(n)*z^n that is analytic and univalent on the unit disk has coefficients a(n) which must satisfy the inequality abs(a(n)) <= sqrt (n) . Considering how long this conjecture has been around, and how many people have worked on this easily stated and understood problem, the proof is remarkably simple and accessible to graduate students in mathematics. This is the kind of result which will probably be refined and smoothed out to eventually be included in undergraduate courses in complex analysis.