is a conformal mapping of a unit disk on any domain and and , then . In more technical terms, "geometric extremality
implies metric extremality." An alternate formulation is that for any schlicht
function
(Krantz 1999, p. 150).
The conjecture had been proven for the first six terms (the cases , 3, and 4 were done by Bieberbach, Lowner, and Garabedian
and Schiffer, 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).
author
result
Bieberbach (1916)
Löwner
(1923)
Garabedian
and Schiffer (1955)
Pederson
(1968), Ozawa (1969)
Pederson
and Schiffer (1972)
de Branges
(1985)
for all
The sum
was an essential tool in de Branges' proof (Koepf 1998, p. 29).
Bieberbach, L. "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln."
Sitzungsber. Preuss. Akad. Wiss., pp. 940-955, 1916.Charzynski,
Z. and Schiffer, M. "A New Proof of the Bieberbach Conjecture for the Fourth
Coefficient." Arch. Rational Mech. Anal.5, 187-193, 1960.de
Branges, L. "A Proof of the Bieberbach Conjecture." Acta Math.154,
137-152, 1985.Duren, P.; Drasin, D.; Bernstein, A.; and Marden, A. The
Bieberbach Conjecture: Proceedings of the Symposium on the Occasion of the Proof.
Providence, RI: Amer. Math. Soc., 1986.Garabedian, P. R. "Inequalities
for the Fifth Coefficient." Comm. Pure Appl. Math.19, 199-214,
1966.Garabedian, P. R.; Ross, G. G.; and Schiffer, M. "On
the Bieberbach Conjecture for Even n." J. Math. Mech.14,
975-989, 1965.Garabedian, R. and Schiffer, M. "A Proof of the Bieberbach
Conjecture for the Fourth Coefficient." J. Rational Mech. Anal.4,
427-465, 1955.Gong, S. The
Bieberbach Conjecture. Providence, RI: Amer. Math. Soc., 1999.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. Monthly95, 689-696, 1988.Koepf, W. "Hypergeometric
Identities." Ch. 2 in Hypergeometric
Summation: An Algorithmic Approach to Summation and Special Function Identities.
Braunschweig, Germany: Vieweg, p. 29, 1998.Korevaar, J. "Ludwig
Bieberbach's Conjecture and its Proof." Amer. Math. Monthly93,
505-513, 1986.Krantz, S. G. "The Bieberbach Conjecture."
§12.1.2 in Handbook
of Complex Variables. Boston, MA: Birkhäuser, pp. 149-150, 1999.Le
Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 53, 1983.Löwner,
K. "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises.
I." Math. Ann.89, 103-121, 1923.Ozawa, M. "On
the Bieberbach Conjecture for the Sixth Coefficient." Kodai Math. Sem. Rep.21,
97-128, 1969.Pederson, R. N. "On Unitary Properties of Grunsky's
Matrix." Arch. Rational Mech. Anal.29, 370-377, 1968.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.Weinstein, L. "The Bieberbach
Conjecture." Internat. Math. Res. Not.5, 61-64, 1991.
th
coefficient in the power
series of a univalent function should be
no greater than 
.
In other words, if
and 
, then 
. In more technical terms, "geometric extremality
implies metric extremality." An alternate formulation is that 
for any schlicht
function 
(Krantz 1999, p. 150).
, 3, and 4 were done by Bieberbach, Lowner, and Garabedian
and Schiffer, 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).
for all 
