3172:
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
940:
2460:. 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.
3429:
3174:
3604:
3027:
2872:
3700:
2287:
336:
679:
2429:
1774:
562:
1962:
268:
4364:
Löwner, C. (1917), "Untersuchungen über die
Verzerrung bei konformen Abbildungen des Einheitskreises /z/ < 1, die durch Funktionen mit nicht verschwindender Ableitung geliefert werden",
1881:
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
487:
1200:
4163:
Grinshpan, Arcadii Z. (2002), "Logarithmic
Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains", in Kuhnau, Reiner (ed.),
1344:
3493:
1684:
1620:
1571:
3087:
2626:
2579:
2528:
1511:
1448:
1405:
1246:
3241:
1819:
819:
808:
3750:
3268:
2905:
1053:
2136:
2667:
2323:
2178:, 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
1079:
963:
376:
178:
145:
3316:
3204:
768:
2060:
1989:
108:
2033:
2458:
2013:
1859:
3452:
3308:
3288:
3107:
2925:
2754:
2727:
2707:
2687:
2481:
2343:
2176:
2156:
2100:
2080:
1879:
1839:
1640:
1591:
1266:
1099:
1010:
990:
733:
702:
589:
416:
396:
3501:
3990:
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
4279:
3164:
verified more of these inequalities by computer for de
Branges (proving the Bieberbach conjecture for the first 30 or so coefficients) and then asked
3756:
4547:
4524:
2933:
2762:
1459:
3896:
Bieberbach, L. (1916), "Ăśber die
Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln",
3615:
4591:
Liu, Xiaosong; Liu, Taishun; Xu, Qinghua (2015). "A proof of a weak version of the
Bieberbach conjecture in several complex variables".
4639:
2195:
4268:
1355:
283:
114:, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that
3039:
that the Milin conjecture (later proved by de
Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture.
4629:
3921:
3879:
17:
2351:
1689:
3137:) about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions (
1888:
196:
3992:, Mathematical Surveys and Monographs, vol. 21, Providence, RI: American Mathematical Society, pp. xvi+218,
597:
4188:
4007:
2434:
Robertson observed that his conjecture is still strong enough to imply the
Bieberbach conjecture, and proved it for
498:
432:
4172:
3160:
De Branges reduced the conjecture to some inequalities for Jacobi polynomials, and verified the first few by hand.
1110:
3812:
4572:
4258:
3789:
4420:
4284:
4127:
4118:
3973:
3861:
1518:
4344:
Littlewood, J.E.; Paley, E. A. C. (1932), "A Proof That An Odd Schlicht Function Has Bounded Coefficients",
3125:. The study of these spaces grew into a sub-field of complex analysis and the spaces have come to be called
4108:
4078:
Garabedian, P. R.; Schiffer, M. (1955). "A Proof of the Bieberbach Conjecture for the Fourth Coefficient".
3154:
3036:
4113:
3965:
3930:
3146:
1271:
1104:
The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function
70:
4250:
4476:
Pederson, Roger N. (December 1968). "A proof of the Bieberbach conjecture for the sixth coefficient".
4497:
Pederson, R.; Schiffer, M. (1972). "A proof of the Bieberbach conjecture for the fifth coefficient".
935:{\displaystyle f_{\alpha }(z)={\frac {z}{(1-\alpha z)^{2}}}=\sum _{n=1}^{\infty }n\alpha ^{n-1}z^{n}}
3457:
1648:
1596:
1530:
3049:
2588:
2541:
2490:
1473:
1410:
1367:
1208:
3209:
1791:
4048:
FitzGerald, Carl; Pommerenke, Christian (1985), "The de Branges theorem on univalent functions",
3708:
3246:
2883:
1781:
705:
4377:
Löwner, C. (1923), "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I",
4519:
4269:
Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality
3970:
Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986)
3424:{\displaystyle \rho _{n}={\frac {\Gamma (2\nu +n+1)}{\Gamma (n+1)}}(\sigma _{n}-\sigma _{n+1})}
1015:
811:
2105:
2637:
2295:
1058:
948:
773:
568:
348:
150:
117:
3183:
738:
4634:
4428:
4321:
4235:
4198:
4156:
4017:
3981:
3958:
3889:
3849:
3760:
2038:
1967:
181:
86:
47:
4573:"The Bieberbach Conjecture by Paul Zorn; Award: Carl B. Allendoerfer; Year of Award: 1987"
4408:
4206:
2018:
8:
3599:{\displaystyle {\frac {F(z)^{\nu }-z^{\nu }}{\nu }}=\sum _{n=1}^{\infty }a_{n}z^{\nu +n}}
3133:) on logarithmic coefficients. This was already known to imply the Robertson conjecture (
2437:
43:
1998:
1844:
4608:
4309:
4144:
4095:
4067:
3988:
Drasin, David; Duren, Peter; Marden, Albert, eds. (1986), "The Bieberbach Conjecture",
3837:
3437:
3293:
3273:
3092:
2910:
2739:
2736:
states that for each schlicht function on the unit disk, and for all positive integers
2712:
2692:
2672:
2466:
2328:
2161:
2141:
2085:
2065:
1864:
1824:
1625:
1576:
1251:
1084:
995:
975:
718:
687:
574:
401:
381:
186:
111:
4180:
3206:
implies the Milin conjecture (and therefore the Bieberbach conjecture). Suppose that
1514:
398:, and it is schlicht, so we cannot find a stricter limit on the absolute value of the
4612:
4301:
4254:
4184:
4087:
4003:
3917:
3875:
3829:
3150:
62:
4538:
2463:
There were several proofs of the Bieberbach conjecture for certain higher values of
4600:
4566:
4562:
4533:
4506:
4485:
4462:
4404:
4394:
4386:
4353:
4333:
4293:
4223:
4202:
4176:
4136:
4125:
Grinshpan, Arcadii Z. (1999), "The Bieberbach conjecture and Milin's functionals",
4057:
4037:
4025:
3993:
3944:
3935:
3865:
3821:
3142:
3126:
31:
4424:
4317:
4231:
4194:
4152:
4013:
3977:
3954:
3913:
3885:
3845:
3161:
3122:
2729:
there can be at most a finite number of exceptions to the Bieberbach conjecture.
1517:. His work was used by most later attempts, and is also applied in the theory of
1463:
4337:
4275:
3905:
3784:
3768:
970:
966:
342:
4604:
4399:
4357:
4227:
4623:
4467:
4450:
4305:
4242:
4091:
4041:
3833:
3807:
3794:
3165:
3118:
1777:
81:
55:
3856:
Baernstein, Albert; Drasin, David; Duren, Peter; et al., eds. (1986),
3968:(1987), "Underlying concepts in the proof of the Bieberbach conjecture",
1992:
180:. That is, we consider a function defined on the open unit disk which is
4436:
Nevanlinna, R. (1921), "Ăśber die konforme Abbildung von Sterngebieten",
4328:
Littlewood, J. E. (1925), "On Inequalities in the Theory of Functions",
4214:
Hayman, W. K. (1955), "The asymptotic behaviour of p-valent functions",
4099:
4510:
4489:
4390:
4313:
4148:
4071:
3998:
3949:
3870:
3860:, Mathematical Surveys and Monographs, vol. 21, Providence, R.I.:
3841:
4168:
58:
51:
4297:
4140:
4062:
3825:
3168:
whether he knew of any similar inequalities. Askey pointed out that
3022:{\displaystyle \log(f(z)/z)=2\sum _{n=1}^{\infty }\gamma _{n}z^{n}.}
2867:{\displaystyle \sum _{k=1}^{n}(n-k+1)(k|\gamma _{k}|^{2}-1/k)\leq 0}
3695:{\displaystyle \sum _{n=1}^{\infty }(\nu +n)\sigma _{n}|a_{n}|^{2}}
1622:
Several authors later reduced the constant in the inequality below
1593:, showing that the Bieberbach conjecture is true up to a factor of
4280:"Ludwig Bieberbach's conjecture and its proof by Louis de Branges"
4249:, Cambridge Tracts in Mathematics, vol. 110 (2nd ed.),
27:
Statement in complex analysis; formerly the Bieberbach conjecture
3810:; Gasper, George (1976), "Positive Jacobi polynomial sums. II",
4577:
Writing Awards, Mathematical Association of America (maa.org)
4028:(1933), "Eine Bemerkung Ăśber Ungerade Schlichte Funktionen",
2282:{\displaystyle \phi (z)=b_{1}z+b_{3}z^{3}+b_{5}z^{5}+\cdots }
2709:
is a Koebe function. In particular this showed that for any
571:: starting with an arbitrary injective holomorphic function
331:{\displaystyle |a_{n}|\leq n\quad {\text{for all }}n\geq 2.}
3855:
3755:
A simplified version of the proof was published in 1985 by
4451:"On the Bieberbach conjecture for the sixth coefficient"
1205:
shows: it is holomorphic on the unit disc and satisfies
3175:
Leningrad Department of Steklov Mathematical Institute
3711:
3618:
3504:
3460:
3440:
3319:
3296:
3276:
3249:
3212:
3186:
3129:. De Branges proved the stronger Milin conjecture (
3095:
3052:
2936:
2913:
2886:
2765:
2742:
2715:
2695:
2675:
2640:
2591:
2544:
2493:
2469:
2440:
2354:
2331:
2298:
2198:
2164:
2144:
2108:
2088:
2068:
2041:
2021:
2001:
1970:
1891:
1867:
1847:
1827:
1794:
1692:
1651:
1628:
1599:
1579:
1533:
1476:
1413:
1370:
1274:
1254:
1211:
1113:
1087:
1061:
1018:
998:
978:
951:
822:
776:
741:
721:
690:
600:
577:
501:
435:
404:
384:
351:
286:
199:
153:
120:
89:
4047:
3764:
2424:{\displaystyle \sum _{k=1}^{n}|b_{2k+1}|^{2}\leq n.}
1885:, who showed there is an odd schlicht function with
1769:{\displaystyle \varphi (z)=z(f(z^{2})/z^{2})^{1/2}}
4077:
3744:
3694:
3598:
3487:
3446:
3423:
3302:
3282:
3262:
3235:
3198:
3180:De Branges proved the following result, which for
3101:
3081:
3021:
2919:
2899:
2866:
2748:
2721:
2701:
2681:
2661:
2620:
2573:
2522:
2484:
2475:
2452:
2423:
2337:
2317:
2292:is an odd schlicht function in the unit disk with
2281:
2170:
2150:
2130:
2094:
2074:
2054:
2027:
2007:
1983:
1956:
1873:
1853:
1833:
1813:
1768:
1678:
1634:
1614:
1585:
1565:
1505:
1442:
1399:
1338:
1260:
1240:
1194:
1093:
1073:
1047:
1004:
984:
957:
934:
802:
762:
727:
696:
673:
583:
556:
481:
410:
390:
370:
330:
262:
172:
139:
102:
3987:
1964:, and that this is the maximum possible value of
1957:{\displaystyle b_{5}=1/2+\exp(-2/3)=1.013\ldots }
263:{\displaystyle f(z)=z+\sum _{n\geq 2}a_{n}z^{n}.}
4621:
4496:
3933:(1985), "A proof of the Bieberbach conjecture",
2582:
2158:). So the limit is always less than or equal to
674:{\displaystyle f(z)={\frac {g(z)-g(0)}{g'(0)}}.}
4343:
3898:Sitzungsber. Preuss. Akad. Wiss. Phys-Math. Kl.
3434:is non-negative, non-increasing, and has limit
1788:) showed that its Taylor coefficients satisfy
1785:
4216:Proceedings of the London Mathematical Society
557:{\displaystyle f(0)=0\ {\text{and}}\ f'(0)=1.}
482:{\displaystyle a_{0}=0\ {\text{and}}\ a_{1}=1}
4525:Bulletin of the American Mathematical Society
4167:, Handbook of Complex Analysis, vol. 1,
704:are of interest because they appear in the
1195:{\displaystyle f(z)=z+z^{2}=(z+1/2)^{2}-1/4}
4499:Archive for Rational Mechanics and Analysis
4478:Archive for Rational Mechanics and Analysis
4417:Univalent functions and orthonormal systems
4023:
3806:
3169:
1882:
810:. A family of schlicht functions are the
4435:
4327:
4080:Journal of Rational Mechanics and Analysis
3964:
3929:
3895:
3138:
3043:
1524:
1455:
1361:
591:defined on the open unit disk and setting
74:
66:
4537:
4517:
4466:
4432:(Translation of the 1971 Russian edition)
4398:
4162:
4124:
4061:
3997:
3948:
3869:
3454:. Then for all Riemann mapping functions
3134:
2669:exists, and has absolute value less than
2179:
4590:
4545:
4520:"A remark on the odd schlicht functions"
4475:
4274:
4106:
3772:
2535:
1458:independently proved the conjecture for
3270:are real numbers for positive integers
2102:is not a Koebe function (for which the
14:
4622:
4376:
4363:
4241:
4213:
3904:
3767:), and an even shorter description by
2631:
1467:
1451:
61:to the complex plane. It was posed by
4448:
4414:
3130:
3112:
3032:
2531:
2035:, and Hayman showed that the numbers
421:
4366:Ber. Verh. Sachs. Ges. Wiss. Leipzig
3910:Functions of One Complex Variable II
1354:A survey of the history is given by
345:(see below) is a function for which
3177:) when de Branges visited in 1984.
1339:{\displaystyle f(-1/2+z)=f(-1/2-z)}
715:is defined as an analytic function
24:
4584:
3765:FitzGerald & Pommerenke (1985)
3705:is achieved by the Koebe function
3635:
3565:
3365:
3336:
2991:
898:
567:This can always be obtained by an
25:
4651:
4640:Conjectures that have been proved
4449:Ozawa, Mitsuru (1 January 1969).
4285:The American Mathematical Monthly
4128:The American Mathematical Monthly
1407:, and stated the conjecture that
735:that is one-to-one and satisfies
190:) with Taylor series of the form
3495:univalent in the unit disk with
2485:Garabedian & Schiffer (1955)
1268:, but it is not injective since
277:. The theorem then states that
4539:10.1090/S0002-9904-1936-06300-7
4245:(1994), "De Branges' Theorem",
3813:American Journal of Mathematics
3157:on exponentiated power series.
2325:then for all positive integers
313:
4567:10.1080/0025570X.1986.11977236
4438:Ofvers. Finska Vet. Soc. Forh.
3733:
3720:
3682:
3666:
3652:
3640:
3518:
3511:
3488:{\displaystyle F(z)=z+\cdots }
3470:
3464:
3418:
3386:
3380:
3368:
3360:
3339:
3069:
3054:
2966:
2955:
2949:
2943:
2855:
2831:
2815:
2808:
2805:
2787:
2608:
2593:
2583:Pederson & Schiffer (1972)
2561:
2546:
2510:
2495:
2402:
2377:
2208:
2202:
1942:
1925:
1776:is an odd schlicht function.
1749:
1730:
1717:
1711:
1702:
1696:
1679:{\displaystyle f(z)=z+\cdots }
1661:
1655:
1615:{\displaystyle e=2.718\ldots }
1566:{\displaystyle |a_{n}|\leq en}
1550:
1535:
1493:
1478:
1430:
1415:
1387:
1372:
1333:
1310:
1301:
1278:
1228:
1213:
1169:
1148:
1123:
1117:
1035:
1020:
867:
851:
839:
833:
791:
785:
751:
745:
662:
656:
643:
637:
628:
622:
610:
604:
545:
539:
511:
505:
303:
288:
209:
203:
13:
1:
4421:American Mathematical Society
4181:10.1016/S1874-5709(02)80012-9
3974:American Mathematical Society
3862:American Mathematical Society
3800:
3082:{\displaystyle |a_{n}|\leq n}
2621:{\displaystyle |a_{5}|\leq 5}
2574:{\displaystyle |a_{6}|\leq 6}
2523:{\displaystyle |a_{4}|\leq 4}
1506:{\displaystyle |a_{3}|\leq 3}
1443:{\displaystyle |a_{n}|\leq n}
1400:{\displaystyle |a_{2}|\leq 2}
1241:{\displaystyle |a_{n}|\leq n}
1101:is a rotated Koebe function.
4630:Theorems in complex analysis
3236:{\displaystyle \nu >-3/2}
1814:{\displaystyle b_{k}\leq 14}
1686:is a schlicht function then
42:, is a theorem that gives a
7:
4548:"The Bieberbach Conjecture"
4114:Encyclopedia of Mathematics
3778:
3745:{\displaystyle z/(1-z)^{2}}
3263:{\displaystyle \sigma _{n}}
2900:{\displaystyle \gamma _{n}}
1012:is a schlicht function and
80:The statement concerns the
50:in order for it to map the
10:
4656:
4455:Kodai Mathematical Journal
4251:Cambridge University Press
1527:, theorem 20) proved that
1349:
273:Such functions are called
4605:10.1007/s11425-015-5016-2
4593:Science China Mathematics
4518:Robertson, M. S. (1936),
4165:Geometric Function Theory
3858:The Bieberbach conjecture
3170:Askey & Gasper (1976)
3117:The proof uses a type of
2634:proved that the limit of
1883:Fekete & Szegő (1933)
1519:Schramm–Loewner evolution
1048:{\displaystyle |a_{n}|=n}
4338:10.1112/plms/s2-23.1.481
4107:Goluzina, E.G. (2001) ,
3155:Lebedev–Milin inequality
3037:Lebedev–Milin inequality
2879:logarithmic coefficients
2131:{\displaystyle b_{2k+1}}
1841:. They conjectured that
69:) and finally proven by
4358:10.1112/jlms/s1-7.3.167
4330:Proc. London Math. Soc.
4266:Koepf, Wolfram (2007),
4228:10.1112/plms/s3-5.3.257
4109:"Bieberbach conjecture"
4050:Trans. Amer. Math. Soc.
3790:Fekete–Szegő inequality
3147:Askey–Gasper inequality
2662:{\displaystyle a_{n}/n}
2318:{\displaystyle b_{1}=1}
2062:have a limit less than
1074:{\displaystyle n\geq 2}
958:{\displaystyle \alpha }
812:rotated Koebe functions
803:{\displaystyle f'(0)=1}
706:Riemann mapping theorem
371:{\displaystyle a_{n}=n}
173:{\displaystyle a_{1}=1}
140:{\displaystyle a_{0}=0}
4468:10.2996/kmj/1138845834
4042:10.1112/jlms/s1-8.2.85
3746:
3696:
3639:
3609:the maximum value of
3600:
3569:
3489:
3448:
3425:
3304:
3284:
3264:
3237:
3200:
3199:{\displaystyle \nu =0}
3141:). His proof uses the
3103:
3083:
3023:
2995:
2921:
2901:
2868:
2786:
2750:
2723:
2703:
2683:
2663:
2622:
2575:
2524:
2477:
2454:
2425:
2375:
2339:
2319:
2283:
2172:
2152:
2132:
2096:
2076:
2056:
2029:
2009:
1985:
1958:
1875:
1855:
1835:
1815:
1770:
1680:
1636:
1616:
1587:
1567:
1507:
1444:
1401:
1340:
1262:
1242:
1196:
1095:
1075:
1049:
1006:
986:
959:
936:
902:
804:
764:
763:{\displaystyle f(0)=0}
729:
698:
675:
585:
558:
483:
412:
392:
372:
332:
264:
174:
141:
104:
4415:Milin, I. M. (1977),
4247:Multivalent functions
3747:
3697:
3619:
3601:
3549:
3490:
3449:
3426:
3305:
3285:
3265:
3238:
3201:
3104:
3084:
3024:
2975:
2922:
2902:
2869:
2766:
2751:
2724:
2704:
2684:
2664:
2623:
2576:
2525:
2478:
2455:
2426:
2355:
2340:
2320:
2284:
2173:
2153:
2133:
2097:
2077:
2057:
2055:{\displaystyle b_{k}}
2030:
2010:
1986:
1984:{\displaystyle b_{5}}
1959:
1876:
1856:
1836:
1816:
1771:
1681:
1637:
1617:
1588:
1568:
1508:
1445:
1402:
1341:
1263:
1243:
1197:
1096:
1076:
1050:
1007:
987:
960:
937:
882:
805:
765:
730:
699:
676:
586:
569:affine transformation
559:
484:
413:
393:
373:
333:
265:
175:
142:
105:
103:{\displaystyle a_{n}}
63:Ludwig Bieberbach
40:Bieberbach conjecture
18:Bieberbach conjecture
4555:Mathematics Magazine
4419:, Providence, R.I.:
4346:J. London Math. Soc.
4175:, pp. 273–332,
4030:J. London Math. Soc.
3972:, Providence, R.I.:
3912:, Berlin, New York:
3864:, pp. xvi+218,
3761:Christian Pommerenke
3709:
3616:
3502:
3458:
3438:
3317:
3294:
3274:
3247:
3210:
3184:
3093:
3050:
2934:
2911:
2884:
2763:
2740:
2713:
2693:
2673:
2638:
2589:
2542:
2491:
2467:
2438:
2352:
2329:
2296:
2196:
2187:Robertson conjecture
2162:
2142:
2106:
2086:
2066:
2039:
2028:{\displaystyle 1.14}
2019:
1999:
1968:
1889:
1865:
1845:
1825:
1792:
1690:
1649:
1626:
1597:
1577:
1531:
1474:
1411:
1368:
1272:
1252:
1209:
1111:
1085:
1059:
1016:
996:
976:
949:
820:
774:
739:
719:
688:
598:
575:
499:
433:
402:
382:
349:
284:
197:
151:
118:
87:
71:Louis de Branges
48:holomorphic function
36:de Branges's theorem
2453:{\displaystyle n=3}
2015:can be replaced by
1861:can be replaced by
426:The normalizations
82:Taylor coefficients
44:necessary condition
4511:10.1007/BF00281531
4490:10.1007/BF00251415
4400:10338.dmlcz/125927
4391:10.1007/BF01448091
4332:, s2-23: 481–519,
3976:, pp. 25–42,
3950:10.1007/BF02392821
3742:
3692:
3596:
3485:
3444:
3421:
3300:
3280:
3260:
3233:
3196:
3151:Jacobi polynomials
3113:De Branges's proof
3099:
3079:
3019:
2917:
2897:
2864:
2746:
2719:
2699:
2679:
2659:
2618:
2571:
2520:
2473:
2450:
2421:
2335:
2315:
2279:
2168:
2148:
2128:
2092:
2072:
2052:
2025:
2008:{\displaystyle 14}
2005:
1995:later showed that
1981:
1954:
1871:
1854:{\displaystyle 14}
1851:
1831:
1811:
1766:
1676:
1632:
1612:
1583:
1563:
1503:
1460:starlike functions
1440:
1397:
1336:
1258:
1238:
1192:
1091:
1071:
1045:
1002:
982:
955:
932:
800:
760:
725:
694:
671:
581:
554:
479:
422:Schlicht functions
408:
388:
368:
328:
260:
236:
170:
137:
112:univalent function
100:
4599:(12): 2531–2540.
4546:Zorn, P. (1986).
3966:de Branges, Louis
3931:de Branges, Louis
3923:978-0-387-94460-9
3881:978-0-8218-1521-2
3544:
3447:{\displaystyle 0}
3384:
3303:{\displaystyle 0}
3283:{\displaystyle n}
3127:de Branges spaces
3102:{\displaystyle n}
3044:de Branges (1987)
3035:showed using the
2920:{\displaystyle f}
2749:{\displaystyle n}
2722:{\displaystyle f}
2702:{\displaystyle f}
2682:{\displaystyle 1}
2476:{\displaystyle n}
2338:{\displaystyle n}
2171:{\displaystyle 1}
2151:{\displaystyle 1}
2095:{\displaystyle f}
2075:{\displaystyle 1}
1874:{\displaystyle 1}
1834:{\displaystyle k}
1635:{\displaystyle e}
1586:{\displaystyle n}
1456:Nevanlinna (1921)
1362:Bieberbach (1916)
1261:{\displaystyle n}
1094:{\displaystyle f}
1005:{\displaystyle f}
985:{\displaystyle 1}
877:
728:{\displaystyle f}
713:schlicht function
697:{\displaystyle g}
666:
584:{\displaystyle g}
530:
526:
522:
462:
458:
454:
411:{\displaystyle n}
391:{\displaystyle n}
317:
221:
16:(Redirected from
4647:
4616:
4580:
4570:
4552:
4542:
4541:
4514:
4493:
4472:
4470:
4445:
4431:
4411:
4402:
4373:
4360:
4340:
4324:
4263:
4238:
4218:, Third Series,
4209:
4159:
4121:
4103:
4074:
4065:
4044:
4020:
4001:
3999:10.1090/surv/021
3984:
3961:
3952:
3936:Acta Mathematica
3926:
3901:
3892:
3873:
3871:10.1090/surv/021
3852:
3751:
3749:
3748:
3743:
3741:
3740:
3719:
3701:
3699:
3698:
3693:
3691:
3690:
3685:
3679:
3678:
3669:
3664:
3663:
3638:
3633:
3605:
3603:
3602:
3597:
3595:
3594:
3579:
3578:
3568:
3563:
3545:
3540:
3539:
3538:
3526:
3525:
3506:
3494:
3492:
3491:
3486:
3453:
3451:
3450:
3445:
3430:
3428:
3427:
3422:
3417:
3416:
3398:
3397:
3385:
3383:
3363:
3334:
3329:
3328:
3309:
3307:
3306:
3301:
3289:
3287:
3286:
3281:
3269:
3267:
3266:
3261:
3259:
3258:
3242:
3240:
3239:
3234:
3229:
3205:
3203:
3202:
3197:
3143:Loewner equation
3123:entire functions
3108:
3106:
3105:
3100:
3088:
3086:
3085:
3080:
3072:
3067:
3066:
3057:
3028:
3026:
3025:
3020:
3015:
3014:
3005:
3004:
2994:
2989:
2962:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2896:
2895:
2873:
2871:
2870:
2865:
2851:
2840:
2839:
2834:
2828:
2827:
2818:
2785:
2780:
2755:
2753:
2752:
2747:
2734:Milin conjecture
2728:
2726:
2725:
2720:
2708:
2706:
2705:
2700:
2688:
2686:
2685:
2680:
2668:
2666:
2665:
2660:
2655:
2650:
2649:
2627:
2625:
2624:
2619:
2611:
2606:
2605:
2596:
2580:
2578:
2577:
2572:
2564:
2559:
2558:
2549:
2529:
2527:
2526:
2521:
2513:
2508:
2507:
2498:
2483:, in particular
2482:
2480:
2479:
2474:
2459:
2457:
2456:
2451:
2430:
2428:
2427:
2422:
2411:
2410:
2405:
2399:
2398:
2380:
2374:
2369:
2344:
2342:
2341:
2336:
2324:
2322:
2321:
2316:
2308:
2307:
2288:
2286:
2285:
2280:
2272:
2271:
2262:
2261:
2249:
2248:
2239:
2238:
2223:
2222:
2180:Robertson (1936)
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2137:
2135:
2134:
2129:
2127:
2126:
2101:
2099:
2098:
2093:
2081:
2079:
2078:
2073:
2061:
2059:
2058:
2053:
2051:
2050:
2034:
2032:
2031:
2026:
2014:
2012:
2011:
2006:
1990:
1988:
1987:
1982:
1980:
1979:
1963:
1961:
1960:
1955:
1938:
1912:
1901:
1900:
1880:
1878:
1877:
1872:
1860:
1858:
1857:
1852:
1840:
1838:
1837:
1832:
1820:
1818:
1817:
1812:
1804:
1803:
1775:
1773:
1772:
1767:
1765:
1764:
1760:
1747:
1746:
1737:
1729:
1728:
1685:
1683:
1682:
1677:
1641:
1639:
1638:
1633:
1621:
1619:
1618:
1613:
1592:
1590:
1589:
1584:
1572:
1570:
1569:
1564:
1553:
1548:
1547:
1538:
1525:Littlewood (1925
1512:
1510:
1509:
1504:
1496:
1491:
1490:
1481:
1449:
1447:
1446:
1441:
1433:
1428:
1427:
1418:
1406:
1404:
1403:
1398:
1390:
1385:
1384:
1375:
1345:
1343:
1342:
1337:
1323:
1291:
1267:
1265:
1264:
1259:
1247:
1245:
1244:
1239:
1231:
1226:
1225:
1216:
1201:
1199:
1198:
1193:
1188:
1177:
1176:
1164:
1144:
1143:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1054:
1052:
1051:
1046:
1038:
1033:
1032:
1023:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
964:
962:
961:
956:
941:
939:
938:
933:
931:
930:
921:
920:
901:
896:
878:
876:
875:
874:
846:
832:
831:
809:
807:
806:
801:
784:
769:
767:
766:
761:
734:
732:
731:
726:
703:
701:
700:
695:
680:
678:
677:
672:
667:
665:
655:
646:
617:
590:
588:
587:
582:
563:
561:
560:
555:
538:
528:
527:
524:
520:
488:
486:
485:
480:
472:
471:
460:
459:
456:
452:
445:
444:
418:th coefficient.
417:
415:
414:
409:
397:
395:
394:
389:
377:
375:
374:
369:
361:
360:
337:
335:
334:
329:
318:
315:
306:
301:
300:
291:
269:
267:
266:
261:
256:
255:
246:
245:
235:
179:
177:
176:
171:
163:
162:
146:
144:
143:
138:
130:
129:
109:
107:
106:
101:
99:
98:
32:complex analysis
21:
4655:
4654:
4650:
4649:
4648:
4646:
4645:
4644:
4620:
4619:
4587:
4585:Further reading
4571:
4550:
4298:10.2307/2323021
4276:Korevaar, Jacob
4261:
4191:
4141:10.2307/2589676
4063:10.2307/2000306
4010:
3924:
3914:Springer-Verlag
3906:Conway, John B.
3882:
3826:10.2307/2373813
3803:
3781:
3773:Korevaar (1986)
3757:Carl FitzGerald
3736:
3732:
3715:
3710:
3707:
3706:
3686:
3681:
3680:
3674:
3670:
3665:
3659:
3655:
3634:
3623:
3617:
3614:
3613:
3584:
3580:
3574:
3570:
3564:
3553:
3534:
3530:
3521:
3517:
3507:
3505:
3503:
3500:
3499:
3459:
3456:
3455:
3439:
3436:
3435:
3406:
3402:
3393:
3389:
3364:
3335:
3333:
3324:
3320:
3318:
3315:
3314:
3295:
3292:
3291:
3275:
3272:
3271:
3254:
3250:
3248:
3245:
3244:
3225:
3211:
3208:
3207:
3185:
3182:
3181:
3162:Walter Gautschi
3139:Bieberbach 1916
3115:
3094:
3091:
3090:
3068:
3062:
3058:
3053:
3051:
3048:
3047:
3010:
3006:
3000:
2996:
2990:
2979:
2958:
2935:
2932:
2931:
2912:
2909:
2908:
2891:
2887:
2885:
2882:
2881:
2847:
2835:
2830:
2829:
2823:
2819:
2814:
2781:
2770:
2764:
2761:
2760:
2741:
2738:
2737:
2714:
2711:
2710:
2694:
2691:
2690:
2674:
2671:
2670:
2651:
2645:
2641:
2639:
2636:
2635:
2607:
2601:
2597:
2592:
2590:
2587:
2586:
2560:
2554:
2550:
2545:
2543:
2540:
2539:
2536:Pederson (1968)
2509:
2503:
2499:
2494:
2492:
2489:
2488:
2468:
2465:
2464:
2439:
2436:
2435:
2406:
2401:
2400:
2385:
2381:
2376:
2370:
2359:
2353:
2350:
2349:
2330:
2327:
2326:
2303:
2299:
2297:
2294:
2293:
2267:
2263:
2257:
2253:
2244:
2240:
2234:
2230:
2218:
2214:
2197:
2194:
2193:
2189:states that if
2163:
2160:
2159:
2143:
2140:
2139:
2113:
2109:
2107:
2104:
2103:
2087:
2084:
2083:
2067:
2064:
2063:
2046:
2042:
2040:
2037:
2036:
2020:
2017:
2016:
2000:
1997:
1996:
1975:
1971:
1969:
1966:
1965:
1934:
1908:
1896:
1892:
1890:
1887:
1886:
1866:
1863:
1862:
1846:
1843:
1842:
1826:
1823:
1822:
1799:
1795:
1793:
1790:
1789:
1756:
1752:
1748:
1742:
1738:
1733:
1724:
1720:
1691:
1688:
1687:
1650:
1647:
1646:
1627:
1624:
1623:
1598:
1595:
1594:
1578:
1575:
1574:
1549:
1543:
1539:
1534:
1532:
1529:
1528:
1515:Löwner equation
1492:
1486:
1482:
1477:
1475:
1472:
1471:
1464:Charles Loewner
1429:
1423:
1419:
1414:
1412:
1409:
1408:
1386:
1380:
1376:
1371:
1369:
1366:
1365:
1352:
1319:
1287:
1273:
1270:
1269:
1253:
1250:
1249:
1227:
1221:
1217:
1212:
1210:
1207:
1206:
1184:
1172:
1168:
1160:
1139:
1135:
1112:
1109:
1108:
1086:
1083:
1082:
1060:
1057:
1056:
1034:
1028:
1024:
1019:
1017:
1014:
1013:
997:
994:
993:
977:
974:
973:
950:
947:
946:
926:
922:
910:
906:
897:
886:
870:
866:
850:
845:
827:
823:
821:
818:
817:
777:
775:
772:
771:
740:
737:
736:
720:
717:
716:
689:
686:
685:
684:Such functions
648:
647:
618:
616:
599:
596:
595:
576:
573:
572:
531:
523:
500:
497:
496:
467:
463:
455:
440:
436:
434:
431:
430:
424:
403:
400:
399:
383:
380:
379:
356:
352:
350:
347:
346:
314:
302:
296:
292:
287:
285:
282:
281:
251:
247:
241:
237:
225:
198:
195:
194:
184:and injective (
158:
154:
152:
149:
148:
125:
121:
119:
116:
115:
94:
90:
88:
85:
84:
28:
23:
22:
15:
12:
11:
5:
4653:
4643:
4642:
4637:
4632:
4618:
4617:
4586:
4583:
4582:
4581:
4561:(3): 131–148.
4543:
4532:(6): 366–370,
4515:
4505:(3): 161–193.
4494:
4484:(5): 331–351.
4473:
4446:
4433:
4412:
4374:
4361:
4352:(3): 167–169,
4341:
4325:
4292:(7): 505–514,
4272:
4264:
4259:
4239:
4222:(3): 257–284,
4211:
4189:
4160:
4135:(3): 203–214,
4122:
4104:
4075:
4045:
4021:
4008:
3985:
3962:
3943:(1): 137–152,
3927:
3922:
3902:
3893:
3880:
3853:
3820:(3): 709–737,
3808:Askey, Richard
3802:
3799:
3798:
3797:
3792:
3787:
3785:Grunsky matrix
3780:
3777:
3769:Jacob Korevaar
3739:
3735:
3731:
3728:
3725:
3722:
3718:
3714:
3703:
3702:
3689:
3684:
3677:
3673:
3668:
3662:
3658:
3654:
3651:
3648:
3645:
3642:
3637:
3632:
3629:
3626:
3622:
3607:
3606:
3593:
3590:
3587:
3583:
3577:
3573:
3567:
3562:
3559:
3556:
3552:
3548:
3543:
3537:
3533:
3529:
3524:
3520:
3516:
3513:
3510:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3443:
3432:
3431:
3420:
3415:
3412:
3409:
3405:
3401:
3396:
3392:
3388:
3382:
3379:
3376:
3373:
3370:
3367:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3332:
3327:
3323:
3310:and such that
3299:
3279:
3257:
3253:
3232:
3228:
3224:
3221:
3218:
3215:
3195:
3192:
3189:
3135:Robertson 1936
3114:
3111:
3098:
3078:
3075:
3071:
3065:
3061:
3056:
3030:
3029:
3018:
3013:
3009:
3003:
2999:
2993:
2988:
2985:
2982:
2978:
2974:
2971:
2968:
2965:
2961:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2916:
2894:
2890:
2875:
2874:
2863:
2860:
2857:
2854:
2850:
2846:
2843:
2838:
2833:
2826:
2822:
2817:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2784:
2779:
2776:
2773:
2769:
2745:
2718:
2698:
2678:
2658:
2654:
2648:
2644:
2617:
2614:
2610:
2604:
2600:
2595:
2570:
2567:
2563:
2557:
2553:
2548:
2519:
2516:
2512:
2506:
2502:
2497:
2472:
2449:
2446:
2443:
2432:
2431:
2420:
2417:
2414:
2409:
2404:
2397:
2394:
2391:
2388:
2384:
2379:
2373:
2368:
2365:
2362:
2358:
2334:
2314:
2311:
2306:
2302:
2290:
2289:
2278:
2275:
2270:
2266:
2260:
2256:
2252:
2247:
2243:
2237:
2233:
2229:
2226:
2221:
2217:
2213:
2210:
2207:
2204:
2201:
2167:
2147:
2125:
2122:
2119:
2116:
2112:
2091:
2071:
2049:
2045:
2024:
2004:
1978:
1974:
1953:
1950:
1947:
1944:
1941:
1937:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1911:
1907:
1904:
1899:
1895:
1870:
1850:
1830:
1810:
1807:
1802:
1798:
1763:
1759:
1755:
1751:
1745:
1741:
1736:
1732:
1727:
1723:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1631:
1611:
1608:
1605:
1602:
1582:
1562:
1559:
1556:
1552:
1546:
1542:
1537:
1502:
1499:
1495:
1489:
1485:
1480:
1439:
1436:
1432:
1426:
1422:
1417:
1396:
1393:
1389:
1383:
1379:
1374:
1351:
1348:
1335:
1332:
1329:
1326:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1290:
1286:
1283:
1280:
1277:
1257:
1237:
1234:
1230:
1224:
1220:
1215:
1203:
1202:
1191:
1187:
1183:
1180:
1175:
1171:
1167:
1163:
1159:
1156:
1153:
1150:
1147:
1142:
1138:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1090:
1070:
1067:
1064:
1044:
1041:
1037:
1031:
1027:
1022:
1001:
981:
971:absolute value
967:complex number
954:
943:
942:
929:
925:
919:
916:
913:
909:
905:
900:
895:
892:
889:
885:
881:
873:
869:
865:
862:
859:
856:
853:
849:
844:
841:
838:
835:
830:
826:
799:
796:
793:
790:
787:
783:
780:
759:
756:
753:
750:
747:
744:
724:
693:
682:
681:
670:
664:
661:
658:
654:
651:
645:
642:
639:
636:
633:
630:
627:
624:
621:
615:
612:
609:
606:
603:
580:
565:
564:
553:
550:
547:
544:
541:
537:
534:
519:
516:
513:
510:
507:
504:
490:
489:
478:
475:
470:
466:
451:
448:
443:
439:
423:
420:
407:
387:
367:
364:
359:
355:
343:Koebe function
339:
338:
327:
324:
321:
312:
309:
305:
299:
295:
290:
271:
270:
259:
254:
250:
244:
240:
234:
231:
228:
224:
220:
217:
214:
211:
208:
205:
202:
169:
166:
161:
157:
136:
133:
128:
124:
97:
93:
52:open unit disk
26:
9:
6:
4:
3:
2:
4652:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4614:
4610:
4606:
4602:
4598:
4594:
4589:
4588:
4578:
4574:
4568:
4564:
4560:
4556:
4549:
4544:
4540:
4535:
4531:
4527:
4526:
4521:
4516:
4512:
4508:
4504:
4500:
4495:
4491:
4487:
4483:
4479:
4474:
4469:
4464:
4461:(1): 97–128.
4460:
4456:
4452:
4447:
4443:
4439:
4434:
4430:
4426:
4422:
4418:
4413:
4410:
4406:
4401:
4396:
4392:
4388:
4384:
4380:
4375:
4371:
4367:
4362:
4359:
4355:
4351:
4347:
4342:
4339:
4335:
4331:
4326:
4323:
4319:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4287:
4286:
4281:
4277:
4273:
4271:
4270:
4265:
4262:
4256:
4252:
4248:
4244:
4243:Hayman, W. K.
4240:
4237:
4233:
4229:
4225:
4221:
4217:
4212:
4208:
4204:
4200:
4196:
4192:
4190:0-444-82845-1
4186:
4182:
4178:
4174:
4173:North-Holland
4170:
4166:
4161:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4129:
4123:
4120:
4116:
4115:
4110:
4105:
4101:
4097:
4093:
4089:
4085:
4081:
4076:
4073:
4069:
4064:
4059:
4055:
4051:
4046:
4043:
4039:
4035:
4031:
4027:
4022:
4019:
4015:
4011:
4009:0-8218-1521-0
4005:
4000:
3995:
3991:
3986:
3983:
3979:
3975:
3971:
3967:
3963:
3960:
3956:
3951:
3946:
3942:
3938:
3937:
3932:
3928:
3925:
3919:
3915:
3911:
3907:
3903:
3899:
3894:
3891:
3887:
3883:
3877:
3872:
3867:
3863:
3859:
3854:
3851:
3847:
3843:
3839:
3835:
3831:
3827:
3823:
3819:
3815:
3814:
3809:
3805:
3804:
3796:
3795:Schwarz lemma
3793:
3791:
3788:
3786:
3783:
3782:
3776:
3774:
3770:
3766:
3762:
3758:
3753:
3737:
3729:
3726:
3723:
3716:
3712:
3687:
3675:
3671:
3660:
3656:
3649:
3646:
3643:
3630:
3627:
3624:
3620:
3612:
3611:
3610:
3591:
3588:
3585:
3581:
3575:
3571:
3560:
3557:
3554:
3550:
3546:
3541:
3535:
3531:
3527:
3522:
3514:
3508:
3498:
3497:
3496:
3482:
3479:
3476:
3473:
3467:
3461:
3441:
3413:
3410:
3407:
3403:
3399:
3394:
3390:
3377:
3374:
3371:
3357:
3354:
3351:
3348:
3345:
3342:
3330:
3325:
3321:
3313:
3312:
3311:
3297:
3277:
3255:
3251:
3230:
3226:
3222:
3219:
3216:
3213:
3193:
3190:
3187:
3178:
3176:
3171:
3167:
3166:Richard Askey
3163:
3158:
3156:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3120:
3119:Hilbert space
3110:
3096:
3076:
3073:
3063:
3059:
3045:
3040:
3038:
3034:
3016:
3011:
3007:
3001:
2997:
2986:
2983:
2980:
2976:
2972:
2969:
2963:
2959:
2952:
2946:
2940:
2937:
2930:
2929:
2928:
2927:are given by
2914:
2892:
2888:
2880:
2861:
2858:
2852:
2848:
2844:
2841:
2836:
2824:
2820:
2811:
2802:
2799:
2796:
2793:
2790:
2782:
2777:
2774:
2771:
2767:
2759:
2758:
2757:
2743:
2735:
2730:
2716:
2696:
2676:
2656:
2652:
2646:
2642:
2633:
2632:Hayman (1955)
2629:
2615:
2612:
2602:
2598:
2584:
2568:
2565:
2555:
2551:
2537:
2533:
2517:
2514:
2504:
2500:
2486:
2470:
2461:
2447:
2444:
2441:
2418:
2415:
2412:
2407:
2395:
2392:
2389:
2386:
2382:
2371:
2366:
2363:
2360:
2356:
2348:
2347:
2346:
2332:
2312:
2309:
2304:
2300:
2276:
2273:
2268:
2264:
2258:
2254:
2250:
2245:
2241:
2235:
2231:
2227:
2224:
2219:
2215:
2211:
2205:
2199:
2192:
2191:
2190:
2188:
2183:
2181:
2165:
2145:
2123:
2120:
2117:
2114:
2110:
2089:
2069:
2047:
2043:
2022:
2002:
1994:
1976:
1972:
1951:
1948:
1945:
1939:
1935:
1931:
1928:
1922:
1919:
1916:
1913:
1909:
1905:
1902:
1897:
1893:
1884:
1868:
1848:
1828:
1808:
1805:
1800:
1796:
1787:
1783:
1780: and
1779:
1761:
1757:
1753:
1743:
1739:
1734:
1725:
1721:
1714:
1708:
1705:
1699:
1693:
1673:
1670:
1667:
1664:
1658:
1652:
1643:
1629:
1609:
1606:
1603:
1600:
1580:
1560:
1557:
1554:
1544:
1540:
1526:
1522:
1520:
1516:
1500:
1497:
1487:
1483:
1469:
1468:Löwner (1923)
1465:
1461:
1457:
1453:
1452:Löwner (1917)
1437:
1434:
1424:
1420:
1394:
1391:
1381:
1377:
1363:
1359:
1357:
1347:
1330:
1327:
1324:
1320:
1316:
1313:
1307:
1304:
1298:
1295:
1292:
1288:
1284:
1281:
1275:
1255:
1235:
1232:
1222:
1218:
1189:
1185:
1181:
1178:
1173:
1165:
1161:
1157:
1154:
1151:
1145:
1140:
1136:
1132:
1129:
1126:
1120:
1114:
1107:
1106:
1105:
1102:
1088:
1068:
1065:
1062:
1042:
1039:
1029:
1025:
999:
979:
972:
968:
952:
927:
923:
917:
914:
911:
907:
903:
893:
890:
887:
883:
879:
871:
863:
860:
857:
854:
847:
842:
836:
828:
824:
816:
815:
814:
813:
797:
794:
788:
781:
778:
757:
754:
748:
742:
722:
714:
709:
707:
691:
668:
659:
652:
649:
640:
634:
631:
625:
619:
613:
607:
601:
594:
593:
592:
578:
570:
551:
548:
542:
535:
532:
517:
514:
508:
502:
495:
494:
493:
476:
473:
468:
464:
449:
446:
441:
437:
429:
428:
427:
419:
405:
385:
365:
362:
357:
353:
344:
325:
322:
319:
316:for all
310:
307:
297:
293:
280:
279:
278:
276:
257:
252:
248:
242:
238:
232:
229:
226:
222:
218:
215:
212:
206:
200:
193:
192:
191:
189:
188:
183:
167:
164:
159:
155:
134:
131:
126:
122:
113:
95:
91:
83:
78:
76:
72:
68:
64:
60:
57:
56:complex plane
53:
49:
45:
41:
37:
33:
19:
4596:
4592:
4576:
4558:
4554:
4529:
4523:
4502:
4498:
4481:
4477:
4458:
4454:
4441:
4437:
4416:
4382:
4378:
4369:
4365:
4349:
4345:
4329:
4289:
4283:
4267:
4246:
4219:
4215:
4164:
4132:
4126:
4112:
4083:
4079:
4053:
4049:
4036:(2): 85–89,
4033:
4029:
4024:Fekete, M.;
3989:
3969:
3940:
3934:
3909:
3897:
3857:
3817:
3811:
3754:
3704:
3608:
3433:
3179:
3159:
3116:
3041:
3033:Milin (1977)
3031:
2878:
2876:
2733:
2731:
2630:
2532:Ozawa (1969)
2462:
2433:
2291:
2186:
2184:
1644:
1523:
1513:, using the
1360:
1356:Koepf (2007)
1353:
1204:
1103:
944:
712:
710:
683:
566:
491:
425:
340:
274:
272:
185:
79:
39:
35:
29:
4635:Conjectures
4385:: 103–121,
4086:: 427–465.
3290:with limit
3153:, and the
1993:Isaak Milin
182:holomorphic
59:injectively
4624:Categories
4409:49.0714.01
4379:Math. Ann.
4260:0521460263
4207:1083.30017
4056:(2): 683,
3801:References
3131:Milin 1977
2877:where the
1782:Littlewood
1055:for some
492:mean that
4613:122080390
4306:0002-9890
4169:Amsterdam
4119:EMS Press
4092:1943-5282
4026:Szegő, G.
3900:: 940–955
3834:0002-9327
3727:−
3657:σ
3644:ν
3636:∞
3621:∑
3586:ν
3566:∞
3551:∑
3542:ν
3536:ν
3528:−
3523:ν
3483:⋯
3404:σ
3400:−
3391:σ
3366:Γ
3346:ν
3337:Γ
3322:ρ
3252:σ
3220:−
3214:ν
3188:ν
3074:≤
2998:γ
2992:∞
2977:∑
2941:
2889:γ
2859:≤
2842:−
2821:γ
2794:−
2768:∑
2613:≤
2566:≤
2515:≤
2413:≤
2357:∑
2277:⋯
2200:ϕ
1952:…
1929:−
1923:
1806:≤
1694:φ
1674:⋯
1610:…
1555:≤
1498:≤
1470:) proved
1435:≤
1392:≤
1328:−
1314:−
1282:−
1233:≤
1179:−
1066:≥
953:α
915:−
908:α
899:∞
884:∑
861:α
858:−
829:α
632:−
323:≥
308:≤
230:≥
223:∑
187:univalent
38:, or the
4372:: 89–106
4278:(1986),
4100:24900366
3908:(1995),
3779:See also
3173:Theory (
3089:for all
3042:Finally
2138:are all
1821:for all
1573:for all
1248:for all
782:′
653:′
536:′
378:for all
275:schlicht
4429:0369684
4322:0856290
4314:2323021
4236:0071536
4199:1966197
4157:1682341
4149:2589676
4072:2000306
4018:0875226
3982:0934213
3959:0772434
3890:0875226
3850:0430358
3842:2373813
3046:proved
2689:unless
2585:proved
2538:proved
2487:proved
1784: (
1462:. Then
1364:proved
1350:History
1081:, then
73: (
65: (
54:of the
4611:
4444:: 1–21
4427:
4407:
4320:
4312:
4304:
4257:
4234:
4205:
4197:
4187:
4155:
4147:
4098:
4090:
4070:
4016:
4006:
3980:
3957:
3920:
3888:
3878:
3848:
3840:
3832:
3149:about
3145:, the
2581:, and
529:
521:
461:
453:
4609:S2CID
4551:(PDF)
4310:JSTOR
4145:JSTOR
4096:JSTOR
4068:JSTOR
3838:JSTOR
1949:1.013
1778:Paley
1607:2.718
1454:and
992:. If
945:with
110:of a
46:on a
4350:s1-7
4302:ISSN
4255:ISBN
4185:ISBN
4088:ISSN
4034:s1-8
4004:ISBN
3918:ISBN
3876:ISBN
3830:ISSN
3759:and
3243:and
3217:>
2732:The
2534:and
2185:The
2023:1.14
1786:1932
770:and
341:The
147:and
75:1985
67:1916
4601:doi
4563:doi
4534:doi
4507:doi
4486:doi
4463:doi
4405:JFM
4395:hdl
4387:doi
4354:doi
4334:doi
4294:doi
4224:doi
4203:Zbl
4177:doi
4137:doi
4133:106
4058:doi
4054:290
4038:doi
3994:doi
3945:doi
3941:154
3866:doi
3822:doi
3775:).
3121:of
2938:log
2907:of
2756:,
2345:,
2082:if
1920:exp
1645:If
969:of
525:and
457:and
77:).
30:In
4626::
4607:.
4597:58
4595:.
4575:.
4559:59
4557:.
4553:.
4530:42
4528:,
4522:,
4503:45
4501:.
4482:31
4480:.
4459:21
4457:.
4453:.
4442:53
4440:,
4425:MR
4423:,
4403:,
4393:,
4383:89
4381:,
4370:69
4368:,
4348:,
4318:MR
4316:,
4308:,
4300:,
4290:93
4288:,
4282:,
4253:,
4232:MR
4230:,
4201:,
4195:MR
4193:,
4183:,
4171::
4153:MR
4151:,
4143:,
4131:,
4117:,
4111:,
4094:.
4082:.
4066:,
4052:,
4032:,
4014:MR
4012:,
4002:,
3978:MR
3955:MR
3953:,
3939:,
3916:,
3886:MR
3884:,
3874:,
3846:MR
3844:,
3836:,
3828:,
3818:98
3816:,
3752:.
3109:.
2628:.
2530:,
2182:.
2003:14
1991:.
1849:14
1809:14
1642:.
1521:.
1450:.
1358:.
1346:.
965:a
711:A
708:.
552:1.
326:2.
34:,
4615:.
4603::
4579:.
4569:.
4565::
4536::
4513:.
4509::
4492:.
4488::
4471:.
4465::
4397::
4389::
4356::
4336::
4296::
4226::
4220:5
4210:.
4179::
4139::
4102:.
4084:4
4060::
4040::
3996::
3947::
3868::
3824::
3771:(
3763:(
3738:2
3734:)
3730:z
3724:1
3721:(
3717:/
3713:z
3688:2
3683:|
3676:n
3672:a
3667:|
3661:n
3653:)
3650:n
3647:+
3641:(
3631:1
3628:=
3625:n
3592:n
3589:+
3582:z
3576:n
3572:a
3561:1
3558:=
3555:n
3547:=
3532:z
3519:)
3515:z
3512:(
3509:F
3480:+
3477:z
3474:=
3471:)
3468:z
3465:(
3462:F
3442:0
3419:)
3414:1
3411:+
3408:n
3395:n
3387:(
3381:)
3378:1
3375:+
3372:n
3369:(
3361:)
3358:1
3355:+
3352:n
3349:+
3343:2
3340:(
3331:=
3326:n
3298:0
3278:n
3256:n
3231:2
3227:/
3223:3
3194:0
3191:=
3097:n
3077:n
3070:|
3064:n
3060:a
3055:|
3017:.
3012:n
3008:z
3002:n
2987:1
2984:=
2981:n
2973:2
2970:=
2967:)
2964:z
2960:/
2956:)
2953:z
2950:(
2947:f
2944:(
2915:f
2893:n
2862:0
2856:)
2853:k
2849:/
2845:1
2837:2
2832:|
2825:k
2816:|
2812:k
2809:(
2806:)
2803:1
2800:+
2797:k
2791:n
2788:(
2783:n
2778:1
2775:=
2772:k
2744:n
2717:f
2697:f
2677:1
2657:n
2653:/
2647:n
2643:a
2616:5
2609:|
2603:5
2599:a
2594:|
2569:6
2562:|
2556:6
2552:a
2547:|
2518:4
2511:|
2505:4
2501:a
2496:|
2471:n
2448:3
2445:=
2442:n
2419:.
2416:n
2408:2
2403:|
2396:1
2393:+
2390:k
2387:2
2383:b
2378:|
2372:n
2367:1
2364:=
2361:k
2333:n
2313:1
2310:=
2305:1
2301:b
2274:+
2269:5
2265:z
2259:5
2255:b
2251:+
2246:3
2242:z
2236:3
2232:b
2228:+
2225:z
2220:1
2216:b
2212:=
2209:)
2206:z
2203:(
2166:1
2146:1
2124:1
2121:+
2118:k
2115:2
2111:b
2090:f
2070:1
2048:k
2044:b
1977:5
1973:b
1946:=
1943:)
1940:3
1936:/
1932:2
1926:(
1917:+
1914:2
1910:/
1906:1
1903:=
1898:5
1894:b
1869:1
1829:k
1801:k
1797:b
1762:2
1758:/
1754:1
1750:)
1744:2
1740:z
1735:/
1731:)
1726:2
1722:z
1718:(
1715:f
1712:(
1709:z
1706:=
1703:)
1700:z
1697:(
1671:+
1668:z
1665:=
1662:)
1659:z
1656:(
1653:f
1630:e
1604:=
1601:e
1581:n
1561:n
1558:e
1551:|
1545:n
1541:a
1536:|
1501:3
1494:|
1488:3
1484:a
1479:|
1466:(
1438:n
1431:|
1425:n
1421:a
1416:|
1395:2
1388:|
1382:2
1378:a
1373:|
1334:)
1331:z
1325:2
1321:/
1317:1
1311:(
1308:f
1305:=
1302:)
1299:z
1296:+
1293:2
1289:/
1285:1
1279:(
1276:f
1256:n
1236:n
1229:|
1223:n
1219:a
1214:|
1190:4
1186:/
1182:1
1174:2
1170:)
1166:2
1162:/
1158:1
1155:+
1152:z
1149:(
1146:=
1141:2
1137:z
1133:+
1130:z
1127:=
1124:)
1121:z
1118:(
1115:f
1089:f
1069:2
1063:n
1043:n
1040:=
1036:|
1030:n
1026:a
1021:|
1000:f
980:1
928:n
924:z
918:1
912:n
904:n
894:1
891:=
888:n
880:=
872:2
868:)
864:z
855:1
852:(
848:z
843:=
840:)
837:z
834:(
825:f
798:1
795:=
792:)
789:0
786:(
779:f
758:0
755:=
752:)
749:0
746:(
743:f
723:f
692:g
669:.
663:)
660:0
657:(
650:g
644:)
641:0
638:(
635:g
629:)
626:z
623:(
620:g
614:=
611:)
608:z
605:(
602:f
579:g
549:=
546:)
543:0
540:(
533:f
518:0
515:=
512:)
509:0
506:(
503:f
477:1
474:=
469:1
465:a
450:0
447:=
442:0
438:a
406:n
386:n
366:n
363:=
358:n
354:a
320:n
311:n
304:|
298:n
294:a
289:|
258:.
253:n
249:z
243:n
239:a
233:2
227:n
219:+
216:z
213:=
210:)
207:z
204:(
201:f
168:1
165:=
160:1
156:a
135:0
132:=
127:0
123:a
96:n
92:a
20:)
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