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比贝尔巴赫猜想

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单叶函数的第 n 个系数在幂级数中应不大于 n。换句话说，如果

f(z)=a_0+a_1z+a_2z^2+...+a_nz^n+...

是共形映射 在任何域上的单位圆盘，且 a_0=0 和 a_1=1，则 |a_n|<=n|a_1|。更专业地说，“几何极值性蕴含度量极值性”。另一种表述是对于任何 schlicht 函数 f，|a_j|<=j (Krantz 1999, p. 150)。

该猜想的前六项已被证明（n=2、3 和 4 的情况分别由 Bieberbach、Lowner 以及 Garabedian 和 Schiffer 完成），已知仅对有限数量的索引为假 (Hayman 1954)，并且对于凸域或对称域为真 (Le Lionnais 1983)。一般情况由 Louis de Branges (1985) 证明。de Branges 证明了 Milin 猜想，该猜想确立了 Robertson 猜想，Robertson 猜想反过来又确立了比贝尔巴赫猜想 (Stewart 1996)。

作者结果
Bieberbach (1916)|a_2|<=2
Löwner (1923)|a_3|<=3
Garabedian and Schiffer (1955)|a_4|<=4
Pederson (1968), Ozawa (1969)|a_6|<=6
Pederson and Schiffer (1972)|a_5|<=5
de Branges (1985)|a_j|<=j 对于所有 j

求和

sum_(j=k)^n(-1)^(k+j)(2j; j-k)(n+j+1; n-j)e^(-jt)

是 de Branges 证明中的一个重要工具 (Koepf 1998, p. 29)。

另请参阅

Milin 猜想, Robertson 猜想, Schlicht 函数, 单叶函数

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参考文献

Bieberbach, L. "Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln。" Sitzungsber. Preuss. Akad. Wiss., pp. 940-955, 1916.Charzynski, Z. and Schiffer, M. "A New Proof of the Bieberbach Conjecture for the Fourth Coefficient。" Arch. Rational Mech. Anal. 5, 187-193, 1960.de Branges, L. "A Proof of the Bieberbach Conjecture。" Acta Math. 154, 137-152, 1985.Duren, P.; Drasin, D.; Bernstein, A.; and Marden, A. The Bieberbach Conjecture: Proceedings of the Symposium on the Occasion of the Proof. Providence, RI: Amer. Math. Soc., 1986.Garabedian, P. R. "Inequalities for the Fifth Coefficient。" Comm. Pure Appl. Math. 19, 199-214, 1966.Garabedian, P. R.; Ross, G. G.; and Schiffer, M. "On the Bieberbach Conjecture for Even n。" J. Math. Mech. 14, 975-989, 1965.Garabedian, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fourth Coefficient。" J. Rational Mech. Anal. 4, 427-465, 1955.Gong, S. The Bieberbach Conjecture. Providence, RI: Amer. Math. Soc., 1999.Hayman, W. K. Multivalent Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1994.Hayman, W. K. and Stewart, F. M. "Real Inequalities with Applications to Function Theory。" Proc. Cambridge Phil. Soc. 50, 250-260, 1954.Kazarinoff, N. D. "Special Functions and the Bieberbach Conjecture。" Amer. Math. Monthly 95, 689-696, 1988.Koepf, W. "Hypergeometric Identities。" Ch. 2 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 29, 1998.Korevaar, J. "Ludwig Bieberbach's Conjecture and its Proof。" Amer. Math. Monthly 93, 505-513, 1986.Krantz, S. G. "The Bieberbach Conjecture。" §12.1.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 149-150, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983.Löwner, K. "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I。" Math. Ann. 89, 103-121, 1923.Ozawa, M. "On the Bieberbach Conjecture for the Sixth Coefficient。" Kodai Math. Sem. Rep. 21, 97-128, 1969.Pederson, R. N. "On Unitary Properties of Grunsky's Matrix。" Arch. Rational Mech. Anal. 29, 370-377, 1968.Pederson, R. N. "A Proof of the Bieberbach Conjecture for the Sixth Coefficient。" Arch. Rational Mech. Anal. 31, 331-351, 1968/1969.Pederson, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fifth Coefficient。" Arch. Rational Mech. Anal. 45, 161-193, 1972.Stewart, I. "The Bieberbach Conjecture。" In From Here to Infinity: A Guide to Today's Mathematics. Oxford, England: Oxford University Press, pp. 164-166, 1996.Weinstein, L. "The Bieberbach Conjecture。" Internat. Math. Res. Not. 5, 61-64, 1991.

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Weisstein, Eric W. "比贝尔巴赫猜想。" 来自 -- 资源。 https://mathworld.net.cn/BieberbachConjecture.html

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