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 Ludwig Bieberbach (1916) and finally proven by Louis de Branges (1985).

The statement concerns the Taylor coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1=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

f(z)=z+\sum _{n\geq 2}a_{n}z^{n}.

Such functions are called schlicht. The theorem then states that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n| \leq n \quad \text{for all }n\geq 2.}

The Koebe function (see below) is a function for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n=n} for all $n$ , and it is schlicht, so we cannot find a stricter limit on the absolute value of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th coefficient.

Schlicht functions

The normalizations

a_{0}=0\ {\text{and}}\ a_{1}=1

mean that

f(0)=0\ {\text{and}}\ f'(0)=1.

This can always be obtained by an affine transformation: starting with an arbitrary injective holomorphic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} defined on the open unit disk and setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=\frac{g(z)-g(0)}{g'(0)}.}

Such functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are of interest because they appear in the Riemann mapping theorem.

A schlicht function is defined as an analytic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} that is one-to-one and satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(0)=1} . A family of schlicht functions are the rotated Koebe functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\alpha(z)=\frac{z}{(1-\alpha z)^2}=\sum_{n=1}^\infty n\alpha^{n-1} z^n}

with $\alpha$ a complex number of absolute value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a schlicht function and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|=n} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geq 2} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a rotated Koebe function.

The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=z+z^2 = (z+1/2)^2 - 1/4}

shows: it is holomorphic on the unit disc and satisfies $|a_{n}|\leq n$ for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , but it is not injective since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(-1/2+z) = f(-1/2-z)} .

History

A survey of the history is given by Koepf (2007).

Bieberbach (1916) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_2|\leq 2} , and stated the conjecture that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|\leq n} . Löwner (1917) and Nevanlinna (1921) independently proved the conjecture for starlike functions. Then Charles Loewner (Löwner (1923)) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_3|\leq 3} , using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution.

Littlewood (1925, theorem 20) proved that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|\leq en} for all $n$ , showing that the Bieberbach conjecture is true up to a factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e=2.718\ldots} Several authors later reduced the constant in the inequality below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=z+\cdots} is a schlicht function then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(z) = z(f(z^2)/z^2)^{1/2}} is an odd schlicht function. Paley and Littlewood (1932) showed that its Taylor coefficients satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_k\leq 14} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . They conjectured that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 14} can be replaced by $1$ as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by Fekete & Szegő (1933), who showed there is an odd schlicht function with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_5=1/2+\exp(-2/3)=1.013\ldots} , and that this is the maximum possible value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_5} . Isaak Milin later showed that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 14} can be replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1.14} , and Hayman showed that the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_k} have a limit less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is not a Koebe function (for which the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{2k+1}} are all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} ). So the limit is always less than or equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} , 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 Robertson (1936).

The Robertson conjecture states that if

\phi (z)=b_{1}z+b_{3}z^{3}+b_{5}z^{5}+\cdots

is an odd schlicht function in the unit disk with $b_{1}=1$ then for all positive integers $n$ ,

\sum _{k=1}^{n}|b_{2k+1}|^{2}\leq n.

Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , in particular Garabedian & Schiffer (1955) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_4|\leq 4} , Ozawa (1969) and Pederson (1968) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_6|\leq 6} , and Pederson & Schiffer (1972) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_5|\leq 5} .

Hayman (1955) proved that the limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n/n} exists, and has absolute value less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} unless Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a Koebe function. In particular this showed that for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^n_{k=1} (n-k+1)(k|\gamma_k|^2-1/k)\le 0}

where the logarithmic coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_n} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log(f(z)/z)=2 \sum^\infty_{n=1}\gamma_nz^n.}

Milin (1977) 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 de Branges (1987) proved Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|\leq n} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

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 (Milin 1977) on logarithmic coefficients. This was already known to imply the Robertson conjecture (Robertson 1936) about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions (Bieberbach 1916). 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 Askey & Gasper (1976) 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu=0} implies the Milin conjecture (and therefore the Bieberbach conjecture). Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu > -3/2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_n} are real numbers for positive integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} with limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} and such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_n=\frac{\Gamma(2\nu+n+1)}{\Gamma(n+1)}(\sigma_n-\sigma_{n+1}) }

is non-negative, non-increasing, and has limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} . Then for all Riemann mapping functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)=z+\cdots} univalent in the unit disk with

{\frac {F(z)^{\nu }-z^{\nu }}{\nu }}=\sum _{n=1}^{\infty }a_{n}z^{\nu +n}

the maximum value of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty(\nu+n)\sigma_n|a_n|^2}

is achieved by the Koebe function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z/(1-z)^2} .

A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke (FitzGerald & Pommerenke (1985)), and an even shorter description by Jacob Korevaar (Korevaar (1986)).

References

Askey, Richard; Gasper, George (1976), "Positive Jacobi polynomial sums. II", American Journal of Mathematics, 98 (3): 709–737, doi:10.2307/2373813, ISSN 0002-9327, JSTOR 2373813, MR 0430358
Baernstein, Albert; Drasin, David; Duren, Peter; et al., eds. (1986), The Bieberbach conjecture, Mathematical Surveys and Monographs, vol. 21, Providence, R.I.: American Mathematical Society, pp. xvi+218, doi:10.1090/surv/021, ISBN 978-0-8218-1521-2, MR 0875226
Bieberbach, L. (1916), "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln", Sitzungsber. Preuss. Akad. Wiss. Phys-Math. Kl.: 940–955
Conway, John B. (1995), Functions of One Complex Variable II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94460-9
de Branges, Louis (1985), "A proof of the Bieberbach conjecture", Acta Mathematica, 154 (1): 137–152, doi:10.1007/BF02392821, MR 0772434
de Branges, Louis (1987), "Underlying concepts in the proof of the Bieberbach conjecture", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, R.I.: American Mathematical Society, pp. 25–42, MR 0934213
Drasin, David; Duren, Peter; Marden, Albert, eds. (1986), "The Bieberbach Conjecture", Proceedings of the symposium on the occasion of the proof of the Bieberbach conjecture held at Purdue University, West Lafayette, Ind., March 11—14, 1985, Mathematical Surveys and Monographs, vol. 21, Providence, RI: American Mathematical Society, pp. xvi+218, doi:10.1090/surv/021, ISBN 0-8218-1521-0, MR 0875226
Fekete, M.; Szegő, G. (1933), "Eine Bemerkung Über Ungerade Schlichte Funktionen", J. London Math. Soc., s1-8 (2): 85–89, doi:10.1112/jlms/s1-8.2.85
FitzGerald, Carl; Pommerenke, Christian (1985), "The de Branges theorem on univalent functions", Trans. Amer. Math. Soc., 290 (2): 683, doi:10.2307/2000306, JSTOR 2000306
Garabedian, P. R.; Schiffer, M. (1955). "A Proof of the Bieberbach Conjecture for the Fourth Coefficient". Journal of Rational Mechanics and Analysis. 4: 427–465. ISSN 1943-5282. JSTOR 24900366.
Goluzina, E.G. (2001) [1994], "Bieberbach conjecture", Encyclopedia of Mathematics, EMS Press
Grinshpan, Arcadii Z. (1999), "The Bieberbach conjecture and Milin's functionals", The American Mathematical Monthly, 106 (3): 203–214, doi:10.2307/2589676, JSTOR 2589676, MR 1682341
Grinshpan, Arcadii Z. (2002), "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains", in Kuhnau, Reiner (ed.), Geometric Function Theory, Handbook of Complex Analysis, vol. 1, Amsterdam: North-Holland, pp. 273–332, doi:10.1016/S1874-5709(02)80012-9, ISBN 0-444-82845-1, MR 1966197, Zbl 1083.30017.
Hayman, W. K. (1955), "The asymptotic behaviour of p-valent functions", Proceedings of the London Mathematical Society, Third Series, 5 (3): 257–284, doi:10.1112/plms/s3-5.3.257, MR 0071536
Hayman, W. K. (1994), "De Branges' Theorem", Multivalent functions, Cambridge Tracts in Mathematics, vol. 110 (2nd ed.), Cambridge University Press, ISBN 0521460263
Koepf, Wolfram (2007), Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality
Korevaar, Jacob (1986), "Ludwig Bieberbach's conjecture and its proof by Louis de Branges", The American Mathematical Monthly, 93 (7): 505–514, doi:10.2307/2323021, ISSN 0002-9890, JSTOR 2323021, MR 0856290
Littlewood, J. E. (1925), "On Inequalities in the Theory of Functions", Proc. London Math. Soc., s2-23: 481–519, doi:10.1112/plms/s2-23.1.481
Littlewood, J.E.; Paley, E. A. C. (1932), "A Proof That An Odd Schlicht Function Has Bounded Coefficients", J. London Math. Soc., s1-7 (3): 167–169, doi:10.1112/jlms/s1-7.3.167
Löwner, C. (1917), "Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises /z/ < 1, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden", Ber. Verh. Sachs. Ges. Wiss. Leipzig, 69: 89–106
Löwner, C. (1923), "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I", Math. Ann., 89: 103–121, doi:10.1007/BF01448091, hdl:10338.dmlcz/125927, JFM 49.0714.01
Milin, I. M. (1977), Univalent functions and orthonormal systems, Providence, R.I.: American Mathematical Society, MR 0369684 (Translation of the 1971 Russian edition)
Nevanlinna, R. (1921), "Über die konforme Abbildung von Sterngebieten", Ofvers. Finska Vet. Soc. Forh., 53: 1–21
Ozawa, Mitsuru (1 January 1969). "On the Bieberbach conjecture for the sixth coefficient". Kodai Mathematical Journal. 21 (1): 97–128. doi:10.2996/kmj/1138845834.
Pederson, Roger N. (December 1968). "A proof of the Bieberbach conjecture for the sixth coefficient". Archive for Rational Mechanics and Analysis. 31 (5): 331–351. doi:10.1007/BF00251415.
Pederson, R.; Schiffer, M. (1972). "A proof of the Bieberbach conjecture for the fifth coefficient". Archive for Rational Mechanics and Analysis. 45 (3): 161–193. doi:10.1007/BF00281531.
Robertson, M. S. (1936), "A remark on the odd schlicht functions", Bulletin of the American Mathematical Society, 42 (6): 366–370, doi:10.1090/S0002-9904-1936-06300-7
Zorn, P. (1986). "The Bieberbach Conjecture" (PDF). Mathematics Magazine. 59 (3): 131–148. doi:10.1080/0025570X.1986.11977236. "The Bieberbach Conjecture by Paul Zorn; Award: Carl B. Allendoerfer; Year of Award: 1987". Writing Awards, Mathematical Association of America (maa.org).

De Branges's theorem

Contents

Schlicht functions

History

De Branges's proof

See also

References

Further reading