Bounds on power series coefficients

Let f be an analytic function on the unit disk with f(0) = 0 and derivative f ′(0) = 1. If f is one-to-one (injective) then this puts a strict limit on the size of the series coefficients.

Let an be the nth coefficient in the power series for f centered at 0. If f is one-to-one then |an| ≤ n for all positive n. Or to put it another way, if |an| > n for any n > 0 then f must take on some value twice.

The statement above was originally known as the Bieberbach conjecture, but is now known as De Branges’ theorem. Ludwig Bieberbach came up with the conjecture in 1916 and Louis de Branges proved it in 1984. Many people between Bieberbach and De Branges made progress toward the final theorem: proving the theorem for some coefficients, proving it under additional hypotheses, proving approximate versions of the theorem, etc.

The function f(z) = z /(1 – z)² shows that the upper bound in De Branges’ theorem is tight. This function is one-to-one on the unit interval, and its nth coefficient is equal to n.