The Bieberbach conjecture and the Milin conjecture

A. I. Aleksandrov ; I. A. Aleksandrov ; L. S. Kopaneva ; G. A. Yuferova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2009), pp. 7-30

Résumé

Some complete solution of the classical Bieberbach conjecture is given (the problem of coefficient for univalent conformal maps). It is used the means accessible to students of high mathematical level.

Détail

Keywords: the parametric method, the coefficients problem
Mots-clés : univalent conformal maps.

Bibliographie

[1] Bieberbach L., “Über die Koeffizienten derjenigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitteln, sitzungsber”, Preuss. Akad. Wiss. Phys. Math. Kl., 138 (1916), 940–955

[2] Löwner K., “Untersuchungen uber schlichte konforme Abbildung des Einheitskreises. I”, Math. Ann., 89 (1923), 103–121 | DOI | MR | Zbl

[3] Littlewood J.E., “On inequalities of the theory of functions”, Proc. London Math. Soc., 23 (1925), 481–519 | DOI | MR | Zbl

[4] Fekete S., “Eine Bemerkung uber ungerade Funktionen”, J. London Math. Soc., 8 (1933), 85–89 | DOI | Zbl

[5] Goluzin G.M., “O koeffitsientakh odnolistnykh funktsii”, Matem. sb., 22(64):3 (1948), 373–380 | MR | Zbl

[6] Bazilevich I.E., “O teoremakh iskazheniya i koeffitsientakh odnolistnykh funktsii”, Matem. sb., 28(70):1 (1951), 147–164 | MR | Zbl

[7] Garabedian P.R., Shiffer M.A., “A proof of the Bieberbach conjecture for the fourth coefficient”, J. Ration. Mech. Anal., 4 (1955), 427–465 | MR | Zbl

[8] Milin M.M., “Otsenka koeffitsientov odnolistnykh funktsii”, DAN SSSR, 160:4 (1965), 769–771 | MR | Zbl

[9] Pederson R.N., “A proof of the Bieberbach conjecture for the sixth coefficient”, Arch. Ration. Mech. and Anal., 31 (1968), 331–351 | DOI | MR | Zbl

[10] Ozawa M., “An elementary proof of the Bieberbach conjecture for the sixth coefficient”, Kodai Math. Semin. Repts., 21 (1969), 129–132 | DOI | MR | Zbl

[11] Pederson R.N., Shiffer M.A., “A proof of the Bieberbach conjecture for the fifth coefficient”, Arch. Ration. Mech. and Anal., 45 (1972), 161–193 | DOI | MR | Zbl

[12] Fitzgerald C.H., “Exponentiation of certain quadratic inequalities for schlicht functions”, Bull. Amer. Math. Soc., 78 (1972), 209–210 | DOI | MR | Zbl

[13] Horoviz D., “A refinement for coefficient estimates of univalent functions”, Proc. Amer. Math. Soc., 54 (1976), 176–178 | DOI | MR

[14] Lowner K., “Untersuchungen uber die Verzerrung bei konformen Abbildung des Einheitskreises $|z|1$, die durch Funktionen mit nicht verschwinden der Ableitung geliefert warden”, Leipziger Berichter, 69 (1917), 89–106 | Zbl

[15] Nevanlinna R., “Über die Konforme Abbildung von Sterngebieten”, Ofvers Finska Vet. Soc. Forh., 53(A):6 (1921)

[16] Dieudonne, “Sur les functions univalentes”, C.R., 192 (1931), 1148–1150 | Zbl

[17] Kaplan W., “Close-to-convex schlicht functions”, Michigan Math. J., 1:2 (1952), 169–185 | DOI | MR | Zbl

[18] Bazilevich I.E., “Oblasti nachalnykh koeffitsientov ogranichennykh odnolistnykh funktsii $r$-kratnoi simmetrii”, Matem. sb., 43:4 (1957), 409–428 | MR | Zbl

[19] Kufarev P.P., “Odno zamechanie ob ekstremalnykh zadachakh teorii odnolistnykh funktsii”, Uchenye zapiski Tomskogo universiteta, 14 (1950), 3–4

[20] Kufarev P.P., “Ob odnom svoistve ekstremalnykh oblastei zadachi koeffitsientov”, DAN SSSR, 97:3 (1954), 391–393 | MR | Zbl

[21] Kufarev P.P., “Odno zamechanie k zadache koeffitsientov”, Uchenye zapiski Tomskogo universiteta, 25 (1955), 15–18 | MR

[22] Kheiman K., Mnogolistnye funktsii, IL, M., 1960

[23] Milin I.M., Odnolistnye funktsii i ortonormirovannye sistemy, Nauka, M., 1971, 256 pp. | MR | Zbl

[24] Bazilevich I.E., “O dispersii koeffitsientov odnolistnykh funktsii”, Matem. sb., 68(110):4 (1965), 549–560 | Zbl

[25] Shirokov N.A., “Teorema regulyarnosti Kheimana”, Zap. nauchn. semin. LOMI AN SSSR, 24, 1972, 182–200 | MR | Zbl

[26] Lebedev N.A., Milin I.M., “Ob odnom neravenstve”, Vestnik Leningradskogo universiteta, 19 (1965), 157–158 | MR | Zbl

[27] Milin I.M., “O koeffitsientakh odnolistnykh funktsii”, DAN SSSR, 176:5 (1967), 1015–1018 | MR | Zbl

[28] Grinshpan A.Z., “Koeffitsienty neravenstva dlya konformnykh otobrazhenii s gomeomorfnym prodolzheniem”, Sibirskii matem. zhurnal, 26:1 (1985), 49–65 | MR | Zbl

[29] Grinshpan A.Z., “Odnolistnye funktsii i regulyarno izmerimye otobrazheniya”, Sibirskii matem. zhurnal, 27:6 (1986), 50–64 | MR | Zbl

[30] Fitzgerald C.N., “Quadratic inequalities and coefficient estimates for schlicht function”, Arch. Rational. Mech. and Anal., 46:5 (1972), 356–368 | DOI | MR | Zbl

[31] Milin I.M., Odnolistnye funktsii i ortonormirovannye sistemy, Nauka, M., 1971 | MR | Zbl

[32] Branges L., A proof of the Bieberbach conjecture, LOMI preprintes, E. 5-84, 21 pp.

[33] Aleksandrov I.A., “Dokazatelstvo L. de Branzha gipotezy I.M. Milina i gipotezy L. Biberbakha”, Sibirskii matem. zhurnal, 28:2 (1987), 7–20 | MR | Zbl

[34] Aleksandrov I.A., Milin I.M., “O gipoteze Biberbakha i logarifmicheskikh koeffitsientakh odnolistnykh funktsii”, Izv. vuzov. Matematika, 1989, no. 8(327), 3–15 | MR

[35] Sadritdinova G.D., “Oblasti s razrezami i svoistva upravlyayuschikh funktsii v uravnenii Levnera”, Tez. dokl. Mezhdunar. konf. po matem. i mekhanike, TGU, Tomsk, 2003

[36] Yuferova G.A., “Uravnenie Levnera i ortogonalnye mnogochleny”, Vestnik TGU, 2007, no. 298, 121–124

[37] Aleksandrov I.A., Aleksandrov A.I., Kasatkina T.V., “Funktsional Milina i polinomy de Branzha”, Aktualnye problemy sovremennoi matematiki, 3, NII MIOO, Novosibirsk, 1997, 13–18 | Zbl

[38] Aleksandrov I.A., “Proizvodyaschaya funktsiya dlya polinomov de Branzha”, Teoriya funktsii i ee primeneniya, Tez. dokl. shkoly-konferentsii (15–22 iyunya 1995 g., g. Kazan), 1995, 4–6

[39] Weinstein L., “The Bieberbach conjecture”, Intern. Math. Res. Notices, 5 (1991), 61–64 | DOI | MR | Zbl

[40] Wilf H., “A footnote on two proof of the Bieberbach conjecture – de Branges Theorem”, Bull. London Math. Soc., 26 (1994), 61–63 | DOI | MR | Zbl

[41] Koef W., Schmersau D., Weinstain's functions and the Askey–Gasper indentity, URL: , 1996 http://www.opus.kobv.de/zib/volltexte/1996/217/ps/SC-96-06.ps

[42] Gradshtein I.S., Ryzhik I.M., Tablitsy integralov, summ, ryadov i proizvedenii, Fiziko-matematicheskaya literatura, M., 1951 | MR

[43] Aleksandrov I.A., Yuferova G.A., “Formula summirovaniya dlya polinomov Chebysheva i ee primenenie”, Vestnik TGU. Matematika i mekhanika, 2009, no. 2(6), 5–13

[44] Uitteker E.T., Vatson G.N., Kurs sovremennogo analiza, ch. 2, GTTN, M.; L., 1934

[45] Aleksandrov A.I., Aleksandrov I.A., “Tochnye otsenki proizvodnykh odnolistnykh funktsii”, Issledovaniya po matematicheskomu analizu i algebre, vyp. 3, TGU, Tomsk, 2001, 12–16

[46] Marty F., “Sur le module des coefficients de Maclaurin d'une function univalente”, Compt rend. Acad. Sci., 198 (1934), 1569–1571

[47] Littlewood J.E., Paley R.E., “A proof the an odd schlicht function has bounded coefficients”, J. London Math. Soc., s1-7:3 (1932), 167–169 | DOI

[48] Robertson M.S., “A mark on the odd schlicht functions”, Bull. Amer. Math. Soc., 42:6 (1936), 366–370 | DOI | MR | Zbl