De Branges's theorem

De Branges's theorem states that every normalized univalent function on the unit disk has Taylor coefficients satisfying |a_n| ≤ n. Its proof established the Bieberbach conjecture.

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Bieberbach conjecture: The conjecture that normalized univalent functions on the unit disk have Taylor coefficients satisfying |a_n| ≤ n. De Branges proved precisely this conjecture.
Ludwig Bieberbach: German mathematician who formulated the coefficient conjecture for normalized univalent functions in 1916. His conjecture defined the problem that de Branges eventually settled.
Taylor series: A power series representation of a function around a point, with coefficients given by its derivatives. The theorem bounds the coefficients in the normalized function's Taylor expansion.
Bieberbach conjecture proof: The 1984 proof by Louis de Branges that established the Bieberbach coefficient bounds. The theorem is the result of this proof, whose length and complexity prompted careful scrutiny.
Univalent function: A holomorphic function that is injective on its domain. Univalence is the defining condition on the functions covered by the theorem.
Milton Schiffer: Mathematician known for contributions to geometric function theory and extremal problems. Schiffer developed influential variational methods for studying the coefficient problem.
Koebe function: The function z/(1−z)^2, an extremal normalized univalent function on the unit disk. Its coefficients attain the theorem's bound for every index.
Bieberbach conjecture for n = 2: The bound |a₂| ≤ 2 for normalized univalent functions, proved by Bieberbach in 1916. This first case initiated the problem that the theorem resolved for all coefficients.
Loewner theory: A method in geometric function theory that studies evolving conformal maps through differential equations. Loewner methods supplied a major route toward coefficient estimates before the final proof.
Charles Loewner: Mathematician who introduced a differential equation for evolving conformal maps. Loewner's method produced sharp early progress on coefficient bounds.