Bieberbach conjecture

A hypothesis enunciated in 1916 by L. Bieberbach [1]: For all functions $ f(z) $ of class $ S $, i.e. functions $ f(z) $ which are regular and univalent in the disc $ | z | < 1 $ and which have the expansion

$$ f(z) = z + \sum _ {n = 2 } ^ \infty c _ {n} z ^ {n} $$

in this disc, one has the estimate $ | c _ {n} | \leq n $, $ n \geq 2 $, and $ | c _ {n} | = n $ only for the Koebe functions

$$ f _ \theta (z) = z (1- e ^ {i \theta } z) ^ {-2} , $$

where $ \theta $ is a real number. Bieberbach proved his conjecture for $ n = 2 $. The problem of finding an accurate estimate of the coefficients for the class $ S $ is a special case of the coefficient problem.

Owing to its simple formulation and profound significance, Bieberbach's conjecture attracted the attention of numerous mathematicians and stimulated the development of different methods in the geometric theory of functions of a complex variable. At the time of writing (1977) the validity of the Bieberbach conjecture had been established for $ n \leq 6 $. It was first proved for $ n = 3 $ in 1923 by K. Loewner who introduced the parametric method (cf. Parametric representation method); other proofs of the estimate $ | c _ {3} | \leq 3 $ subsequently appeared, and were based on variational methods, on parametric methods and on the method of the extremal metric. The Bieberbach conjecture for $ n = 4 $ was first proved in 1955 by the simultaneous use of variational and parametric methods. In 1960 the estimate $ | c _ {4} | \leq 4 $ was obtained much more simply, with the aid of Grunsky's univalence condition. This estimate was also obtained by variational methods, and with the aid of geometric reasoning; it was once more obtained by using the Grunsky inequalities in matrix form. The validity of Bieberbach's conjecture for $ n = 6 $ was demonstrated in 1968 with the aid of the Grunsky inequalities; it was proved for $ n = 5 $ in 1972 by variational methods. Among other results obtained in trying to prove the Bieberbach conjecture, the following are worthy of note.

W.K. Hayman [4] obtained a number of results on the asymptotic behaviour of the coefficients of functions which are $ p $- sheeted in $ | z | < 1 $, in particular for the class $ S $. He proved that the limit

$$ \lim\limits _ {n \rightarrow \infty } \ \frac{| c _ {n} | }{n} = \alpha _ {f} $$

exists and that $ \alpha _ {f} \leq 1 $ with an equality sign only for the Koebe functions.

A number of studies are available on the local Bieberbach conjecture, i.e. the conjecture that the Koebe function gives $ \max | c _ {n} | $ at least for those functions in the class $ S $ which are close to it in the relevant topology (cf. Univalent function). It was found that for any $ n = 3, 4 \dots $ there exists a sufficiently small $ \epsilon _ {n} > 0 $ such that for a function $ f(z) \in S $ which satisfies the condition $ | c _ {2} - 2 | < \epsilon _ {n} $ the estimate $ \mathop{\rm Re} c _ {n} \leq n $ is valid, and $ \mathop{\rm Re} c _ {n} = n $ only for $ f _ {0} (z) $.

An estimate of the coefficients for all $ n $, accurate as regards the order of dependence on $ n $, was first obtained in 1925 by J.E. Littlewood by reducing the estimate for the coefficients to an estimate for the (average) integral of the modulus. More accurate estimates were obtained in 1951 by I.E. Bazilevich ( $ | c _ {n} | < (e / 2) n + \textrm{ const } $), and in 1965 by I.M. Milin ( $ | c _ {n} | < 1.243 n $, $ n \geq 2 $).

The best estimates to date (1977) were obtained in 1972 [7]:

$$ | c _ {n} | < \sqrt { \frac{7}{6} } n < \ 1.081 n,\ \ n \geq 2, $$

and in 1976 [10]:

$$ | c _ {n} | < \ 1.0691 n,\ \ n \geq 2. $$

For reviews of studies on this subject see [2], [3], [9].

References

[1]	L. Bieberbach, "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln" Sitzungsber. Preuss. Akad. Wiss. Phys-Math. Kl. (1916) pp. 940–955
[2]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3]	I.M. Milin, "Univalent functions and orthonormal systems" , Transl. Math. Monogr. , 49 , Amer. Math. Soc. (1977) (Translated from Russian)
[4]	W.K. Hayman, "The asymptotic behaviour of $p$-valent functions" Proc. London Math. Soc. (3) , 5 (1955) pp. 257–284
[5a]	M. Ozawa, "On the Bieberbach conjecture for the sixth coefficient" Kodai Math. Sem. Rep. , 21 (1969) pp. 97–128
[5b]	M. Ozawa, "An elementary proof of the Bieberbach conjecture for the sixth coefficient" Kodai Math. Sem. Rep. , 21 (1969) pp. 129–132
[6]	R.N. Pederson, M.M. Schiffer, "A proof of the Bieberbach conjecture for the fifth coefficient" Arch. Rat. Mech. and Anal. , 45 (1972) pp. 161–193
[7]	C.H. FitzGerald, "Quadratic inequalities and coefficient estimates for the fifth coefficient" Arch. Rat. Mech. and Anal. , 46 (1972) pp. 356–368
[8]	N.A. Shirokov, "On a regularity theorem of Hayman" J. Soviet Math. , 2 : 6 (1974) pp. 693–710 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. , 24 (1972) pp. 182–200
[9]	I.E. Bazilevich, , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) pp. 444–472 (In Russian)
[10]	D. Horowitz, "A refinement estimate for univalent functions" Proc. Amer. Math. Soc. , 54 (1976) pp. 176–178

Comments

Actually Bieberbach proved $ | c _ {2} | \leq 2 $ and then asked in a footnote if perhaps generally $ | c _ {n} | \leq n $, cf. [1].

In the laborious progress on the Bieberbach conjecture from 1950 until 1975, M. Schiffer's variational methods played a major role. For a detailed history, see Duren's book [a3].

The case $ n = 4 $ was proved by P.R. Garabedian and M.M. Schiffer in 1955.

In 1984 the Bieberbach conjecture was established in complete generality by the French-born U.S. mathematician Louis de Branges [a1], [a2]. In fact, he proved a remarkable hypothesis of the Soviet mathematicians N.A. Lebedev and I.M. Milin which was known to be even stronger than the Bieberbach conjecture, cf. [3], [a3]. The Lebedev–Milin hypothesis asserted that for $ n = 2, 3 \dots $

$$ \tag{* } \Omega _ {n} (0) = \ \sum _ {k = 1 } ^ { {n } - 1 } \left \{ k | \gamma _ {k} | ^ {2} - { \frac{1}{k} } \right \} (n - k) \leq 0. $$

Here the numbers $ \gamma _ {k} $ are the logarithmic coefficients of $ f (z) $, defined by

$$ { \frac{1}{2} } \mathop{\rm log} \ { \frac{f (z) }{z} } = \ \sum _ { k=1 } ^ \infty \gamma _ {k} z ^ {k} . $$

Every function $ f (z) $ in $ S $ is the starting point $ f (z, 0) $ of a Loewner parametric family $ \{ f (z, t) \} $, $ t \geq 0 $, of univalent functions. The latter map the disc onto a continuously-increasing family of domains with limit $ \mathbf C $. Normalized so that $ f (z, t)/e ^ {t} $ is in $ S $, the functions of a Loewner family satisfy Loewner's partial differential equation

$$ \frac{\partial f }{\partial t } = \ z \frac{\partial f }{\partial z } p (z, t), $$

where $ \mathop{\rm Re} p (z, t) > 0 $[a4], [a3]. De Branges introduced functionals $ \Omega _ {n} (t) $ corresponding to the logarithmic coefficients $ \gamma _ {k} (t) $ for $ f (z, t)/e ^ {t} $, replacing the weight factors $ n - k $ in (*) by undetermined weights $ \sigma _ {nk} (t) $ with initial value $ n - k $. By ingenious use of Loewner's differential equation he could show that the weights $ \sigma _ {nk} (t) $ may be chosen such that $ \Omega _ {n} ^ \prime (t) \geq 0 $ for all $ t \geq 0 $ and all $ n $. The proof also used a sophisticated positivity result for hypergeometric functions due to R. Askey and G. Gasper. Since $ \Omega _ {n} (t) \rightarrow 0 $ as $ t \rightarrow \infty $ it followed that $ \Omega _ {n} (0) \leq 0 $. The latter inequality established the Lebedev–Milin hypothesis (*) and thereby also Bieberbach's conjecture $ | c _ {n} | \leq n $. The case of equality was dealt with both by de Branges [a2] and by C.H. FitzGerald and C. Pommerenke [a5].

References

[a1]	L. de Branges, "A proof of the Bieberbach conjecture" Preprint E-5–84. Steklov Math. Inst. Leningrad (1984) pp. 1–21
[a2]	L. de Branges, "A proof of the Bieberbach conjecture" Acta Math. , 154 (1985) pp. 137–152
[a3]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a4]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a5]	C.H. FitzGerald, C. Pommerenke, "The de Branges theorem on univalent functions" Trans. Amer. Math. Soc. , 290 (1985) pp. 683–690