some inequalities for Jacobi polynomials, and verified the first few by hand. Walter Gautschi verified more of these inequalities by computer for de Branges (proving the Bieberbach conjecture for the first 30 or so coefficients) and then asked Richard Askey whether he knew of any similar inequalities. Askey pointed out that had proved the necessary inequalities eight years before, which allowed de Branges to complete his proof. The first version was very long and had some minor mistakes which caused some skepticism about it, but these were corrected with the help of members of the Leningrad seminar on Geometric Function Theory (Leningrad Department of Steklov Mathematical Institute) when de Branges visited in 1984. De Branges proved the following result, which for implies the Milin conjecture (and therefore the Bieberbach conjecture). Suppose that and are real numbers for positive integers with limit and such that is non-negative, non-increasing, and has limit . Then for all Riemann mapping functions univalent in the unit disk with the maximinum value of is achieved by the Koebe function . A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke (), and an even shorter description by Jacob Korevaar (). See also Grunsky matrix Fekete–Szegő inequality Schwarz lemma References . Koepf, Wolfram (2007), Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality (Translation of the 1971 Russian edition) Further reading Theorems in complex analysis Conjectures Conjectures that have been proved