easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by , who showed there is an odd schlicht function with , and that this is the maximum possible value of . Isaak Milin later showed that can be replaced by , and Hayman showed that the numbers have a limit less than if is not a Koebe function (for which the are all ). So the limit is always less than or equal to , meaning that Littlewood and Paley's conjecture is true for all but a finite number of coefficients. A weaker form of Littlewood and Paley's conjecture was found by . The Robertson conjecture states that if is an odd schlicht function in the unit disk with then for all positive integers , Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for . This conjecture introduced the key idea of bounding various quadratic functions of the coefficients rather than the coefficients themselves, which is equivalent to bounding norms of elements in certain Hilbert spaces of schlicht functions. There were several proofs of the Bieberbach conjecture for certain higher values of , in particular proved , and proved , and proved . proved that the limit of exists, and has absolute value less than unless is a Koebe function. In particular this showed that for any there can be at most a finite number of exceptions to the Bieberbach conjecture. The Milin conjecture states that for each schlicht function on the unit disk, and for all positive integers , where the logarithmic coefficients of are given by showed using the Lebedev–Milin inequality that the Milin conjecture (later proved by de Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture. Finally proved for all . De Branges's proof The proof uses a type of Hilbert space of entire functions. The study of these spaces grew into a sub-field of complex analysis and the spaces have come to be called de Branges spaces. De Branges proved the stronger Milin conjecture on logarithmic coefficients. This was already known to imply the Robertson conjecture about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions . His proof uses the Loewner equation, the Askey–Gasper inequality about Jacobi polynomials, and the Lebedev–Milin inequality on exponentiated power series. De Branges reduced the conjecture to