Ludwig Bieberbach's Conjecture and its Proof by Louis de Branges

Abstract

Summary1984 has been an exciting year for complex analysis. It even brought strong rumors that the Riemann hypothesis had been proved, but so far, the rumor has not been confirmed. However, we know for sure that the difficult Bieberbach conjecture has been settled this year. As many of you know, this famous conjecture of 1916 concerns the class S of normalized injective holomorphic functions. That class consists of the 1–1 holomorphic functions from the unit disc U into the complex plane C with a power series of the formf(z)=z + a2z2 + … +anzn + …,|z|< 1.The conjecture asserts that |an| ⩽ n for very f in S and every n. Louis de Branges has proved this conjecture as well as some stronger conjectures for the class S.Each of the following items has played an essential role in the proof: (i) Löwner's partial differential equation for so-called Löwner chains {f,(z)} of injective holomorphic functions from U to C.(ii) The observations of Lebedev and I. M. Milin, especially their inspired conjecture for the so-called logarithmic coefficients of f in S, that is, the coefficients in the expansion for a branch of log {f(z)/z}.(iii) De Branges' striking breakthrough, namely, the creation of a functional associated with the Lebedev-Milin conjecture which varies monotonically along Löwner chains.(iv) De Branges' introduction and solution of a system of differential equations which he devised to make the functional manageable.(v) A positivity result for hypergeometric functions which is a tool in establishing the monotonicity of the functional.Of the above, (i) dates back to 1923, while (ii) and (v) are relatively recent. The Lebedev-Milin observations date from the years 1965–1970 and became well known in the West only around 1977. The hypergeometric functions result occurs in work of Askey and Gasper of 1976.