De Branges's theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients "a" of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that "a" = 0 and "a" = 1. That is, we consider a  function defined on the open unit disk which is holomorphic and injective ("univalent") with Taylor series of the form Such functions are called "schlicht". The theorem then states that The Koebe function (see below) is a function in which "a" = "n" for all "n", and it is schlicht, so we cannot find a stricter limit on the absolute value of the "n"th coefficient. Schlicht functions. The normalizations mean that This can always be obtained by an affine transformation: starting with an arbitrary injective holomorphic function "g" defined on the open unit disk and setting Such functions "g" are  of interest because they appear in the Riemann mapping theorem. A schlicht function is defined as an analytic function "f" that is one-to-one and satisfies "f"(0) = 0 and "f" '(0) = 1. A family of schlicht functions are the rotated Koebe functions with α a complex number of absolute value 1. If "f" is a schlicht function and |"a"| = "n" for some "n" ≥ 2, then "f" is a rotated Koebe function. The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function shows: it is holomorphic on the  unit disc and satisfies |"a"|≤"n" for all "n", but it is not injective since "f"(−1/2 + "z") = "f"(−1/2 − "z"). History. A survey of the history is given by Koepf (2007). Then Charles Loewner () proved |"a"| ≤ 3, using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution. If "f"("z") = "z" + ... is a schlicht function then φ("z") = "f"("z") is an odd schlicht function. The Robertson conjecture states that if is an odd schlicht function in the unit disk with "b"=1 then for all  positive integers "n", Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for "n" = 3. 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 "n", in particular proved |"a"| ≤ 4, and proved |"a"| ≤ 6, and proved |"a"| ≤ 5. The Milin conjecture states that for each schlicht function on the unit  disk, and for all positive integers "n", where the logarithmic coefficients γ of "f" are given by Finally proved |"a"| ≤ "n" for all "n". De Branges's proof. The proof uses a type of Hilbert spaces 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 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 ν = 0 implies the Milin conjecture (and therefore the Bieberbach conjecture). Suppose that ν > −3/2 and σ are real numbers for positive integers "n" with limit 0 and such that is non-negative, non-increasing, and has limit 0. Then for all Riemann mapping functions "F"("z") = "z" +  ... univalent in the unit disk with the maximinum value of is achieved by the Koebe function "z"/(1 − "z"). A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke (), and an even shorter description by Jacob Korevaar ().