Bieberbach Conjecture

The $n$th Coefficient in the Power series of a Univalent Function should be no greater than $n$. In other words, if

$$f(z)=a_0+a_1z+a_2 z^2+\ldots+a_nz^n+\ldots$$

is a conformal transformation of a unit disk on any domain, then $\vert a_n\vert\leq n\vert a_1\vert$. In more technical terms, ``geometric extremality implies metric extremality.'' The conjecture had been proven for the first six terms (the cases $n=2$, 3, and 4 were done by Bieberbach, Lowner, and Shiffer and Garbedjian, respectively), was known to be false for only a finite number of indices (Hayman 1954), and true for a convex or symmetric domain (Le Lionnais 1983). The general case was proved by Louis de Branges (1985). De Branges proved the Milin Conjecture, which established the Robertson Conjecture, which in turn established the Bieberbach conjecture (Stewart 1996).

References

de Branges, L. ``A Proof of the Bieberbach Conjecture.'' Acta Math. 154, 137-152, 1985.

Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1994.

Hayman, W. K. and Stewart, F. M. ``Real Inequalities with Applications to Function Theory.'' Proc. Cambridge Phil. Soc. 50, 250-260, 1954.

Kazarinoff, N. D. ``Special Functions and the Bieberbach Conjecture.'' Amer. Math. Monthly 95, 689-696, 1988.

Korevaar, J. ``Ludwig Bieberbach's Conjecture and its Proof.'' Amer. Math. Monthly 93, 505-513, 1986.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983.

Pederson, R. N. ``A Proof of the Bieberbach Conjecture for the Sixth Coefficient.'' Arch. Rational Mech. Anal. 31, 331-351, 1968/1969.

Pederson, R. and Schiffer, M. ``A Proof of the Bieberbach Conjecture for the Fifth Coefficient.'' Arch. Rational Mech. Anal. 45, 161-193, 1972.

Stewart, I. ``The Bieberbach Conjecture.'' In From Here to Infinity: A Guide to Today's Mathematics. Oxford, England: Oxford University Press, pp. 164-166, 1996.