43:
610:
1699:
994:
1537:
342:
1826:
694:
1694:{\displaystyle 0.4332\approx {\frac {\sqrt {3}}{4}}+2\times 10^{-14}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}\approx 0.47186,}
605:{\displaystyle |f''(z)|=\left|{\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f'(w)}{(w-z)^{2}}}\,\mathrm {d} w\right|\leq {\frac {1}{2\pi }}\cdot 2\pi r\sup _{w=\gamma (t)}{\frac {|f'(w)|}{|w-z|^{2}}}\leq {\frac {2}{r}},}
1729:
1894:
166:
989:{\displaystyle |(f(z)-w)-(z-w)|={\frac {1}{2}}|z|^{2}|f''(tz)|\leq {\frac {|z|^{2}}{1-t|z|}}\leq {\frac {|z|^{2}}{1-|z|}}={\frac {1}{6}}<|z|-|w|\leq |z-w|.}
60:
2114:
1835:
2119:
2003:
1821:{\displaystyle 0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}=0.543258965342...\,\!}
96:
329:
95:. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after
1956:
Baernstein, Albert II; Vinson, Jade P. (1998). "Local minimality results related to the Bloch and Landau constants".
1927:
1497:
1867:
1004:
1905:
1842:
In their paper, Ahlfors and
Grunsky conjectured that their upper bounds are actually the true values of
293:
88:
639:
1423:
In the proof of Landau's
Theorem above, Rouché's theorem implies that not only can we find a disk
2049:
2020:
1944:
31:
121:
2087:
2068:
1989:
1516:. The lower bound 1/72 in Bloch's theorem is not the best possible. Bloch's theorem tells us
2041:
2012:
1981:
1936:
76:
248:
2032:
Landau, Edmund (1929), "Über die
Blochsche Konstante und zwei verwandte Weltkonstanten",
1967:"Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation"
1922:
1705:
1347:
Repeating this argument, we either find a disk of radius at least 1/24 in the range of
172:
17:
1708:. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to
2108:
2053:
2024:
1966:
1948:
236:
2090:
2071:
1918:
1709:
80:
1993:
2095:
2076:
1854:
1474:
92:
2045:
2016:
1940:
1985:
292:
Bloch's theorem corresponds to
Valiron's theorem via the so-called
2001:
Chen, Huaihui; Gauthier, Paul M. (1996). "On Bloch's constant".
171:
Bloch's
Theorem states that there is a disk S ⊂ D on which f is
187:
is a holomorphic function in the unit disk with the property |
36:
198:
be the radius of the largest disk contained in the image of
1830:
261:
is a non-constant entire function then there exist disks
247:
Bloch's theorem was inspired by the following theorem of
1723:. Its exact value is also unknown, but it is known that
265:
of arbitrarily large radius and analytic functions φ in
54:
1870:
1732:
1540:
1351:, proving the theorem, or find an infinite sequence (
697:
345:
124:
51:
A request that this article title be changed to
1138:
be an analytic function in the unit disk such that |
1857:holomorphic functions on the unit disk, a constant
1888:
1820:
1693:
988:
604:
160:
1817:
1274:| < 1/8, then by the first case, the range of
1183:| < 1/4, then by the first case, the range of
619:where γ is the counterclockwise circle of radius
205:Landau's theorem states that there is a constant
1960:. Ann Arbor: Springer, New York. pp. 55–89.
501:
1974:Annales de la Faculté des Sciences de Toulouse
8:
111:be a holomorphic function in the unit disk |
68:this article until the discussion is closed.
1861:can similarly be defined. It is known that
1496:, so its inverse φ is also analytic by the
175:and f(S) contains a disk with radius 1/72.
1520: ≥ 1/72, but the exact value of
1011:contains the disk of radius 1/6 around 0.
1869:
1816:
1794:
1773:
1754:
1745:
1731:
1666:
1645:
1626:
1617:
1594:
1590:
1572:
1547:
1539:
1438:inside the unit disk such that for every
978:
964:
956:
948:
940:
932:
919:
908:
900:
886:
881:
872:
869:
858:
850:
833:
828:
819:
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808:
783:
777:
772:
763:
753:
745:
698:
696:
589:
577:
572:
557:
550:
528:
525:
504:
473:
457:
456:
447:
409:
403:
381:
368:
346:
344:
147:
125:
123:
1925:(1937). "Über die Blochsche Konstante".
1906:Table of selected mathematical constants
1427:of radius at least 1/24 in the range of
1715:The similarly defined optimal constant
1503:
1403:In the latter case the sequence is in
30:For the quantum physics theorem, see
7:
1958:Quasiconformal mappings and analysis
1889:{\displaystyle 0.5<A\leq 0.7853}
646:in the unit disk, there exists 0 ≤
1788:
1767:
1748:
1719:in Landau's theorem is called the
1660:
1639:
1620:
1074:) ≠ 0, the case above applied to (
458:
55:Bloch's theorem (complex analysis)
25:
1431:, but there is also a small disk
1029:) denote the open disk of radius
224:is greater than Bloch's constant
2115:Unsolved problems in mathematics
41:
1104:(0)) implies that the range of
2004:Journal d'Analyse Mathématique
1804:
1791:
1783:
1770:
1764:
1751:
1676:
1663:
1655:
1642:
1636:
1623:
1504:Bloch's and Landau's constants
979:
965:
957:
949:
941:
933:
909:
901:
882:
873:
859:
851:
829:
820:
809:
805:
796:
784:
773:
764:
746:
742:
730:
724:
715:
709:
703:
699:
573:
558:
551:
547:
541:
529:
520:
514:
444:
431:
426:
420:
369:
365:
359:
347:
148:
144:
138:
126:
1:
309:We first prove the case when
2120:Theorems in complex analysis
1415:(0, 1/2), a contradiction.
235:This theorem is named after
1278:contains a disk of radius |
1187:contains a disk of radius |
1040:. For an analytic function
87:describes the behaviour of
2136:
1527:The best known bounds for
1134:For the general case, let
325:)| ≤ 2 in the unit disk.
209:defined as the infimum of
115:| ≤ 1 for which
29:
2034:Mathematische Zeitschrift
1928:Mathematische Zeitschrift
330:Cauchy's integral formula
304:
161:{\displaystyle |f'(0)|=1}
1498:inverse function theorem
1300:Otherwise, there exists
1198:Otherwise, there exists
216:over all such functions
1477:analytic function from
1418:
242:
178:
18:Landau's constants
1919:Ahlfors, Lars Valerian
1890:
1822:
1695:
990:
606:
162:
1965:Bloch, André (1925).
1891:
1823:
1696:
991:
607:
163:
89:holomorphic functions
1868:
1730:
1538:
695:
684:| < 1/6, we have
343:
122:
27:Mathematical theorem
1524:is still unknown.
191:(0)| = 1, then let
2088:Weisstein, Eric W.
2069:Weisstein, Eric W.
2046:10.1007/BF01187791
2017:10.1007/BF02787110
1941:10.1007/BF01160101
1886:
1818:
1691:
1446:there is a unique
986:
602:
524:
332:, we have a bound
158:
2091:"Landau Constant"
1814:0.543258965342...
1808:
1802:
1781:
1762:
1721:Landau's constant
1680:
1674:
1653:
1634:
1612:
1611:
1599:
1557:
1553:
927:
914:
864:
761:
597:
584:
500:
486:
454:
397:
294:Bloch's Principle
243:Valiron's theorem
73:
72:
16:(Redirected from
2127:
2101:
2100:
2082:
2081:
2072:"Bloch Constant"
2056:
2028:
1997:
1986:10.5802/afst.335
1971:
1961:
1952:
1895:
1893:
1892:
1887:
1833:
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1825:
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1819:
1809:
1807:
1803:
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1782:
1774:
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1700:
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1613:
1607:
1600:
1595:
1592:
1591:
1580:
1579:
1558:
1549:
1548:
1514:Bloch's constant
1411:is unbounded in
1375:| < 1/2 and |
1321:| < 1/8 and |
1219:| < 1/4 and |
1005:Rouché's theorem
995:
993:
992:
987:
982:
968:
960:
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936:
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640:Taylor's theorem
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305:Landau's theorem
179:Landau's theorem
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165:
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159:
151:
137:
129:
77:complex analysis
61:under discussion
57:
45:
44:
37:
21:
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2134:
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2125:
2124:
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2104:
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2085:
2067:
2066:
2063:
2031:
2000:
1969:
1964:
1955:
1923:Grunsky, Helmut
1917:
1914:
1902:
1866:
1865:
1829:
1787:
1747:
1728:
1727:
1704:where Γ is the
1659:
1619:
1593:
1568:
1536:
1535:
1531:at present are
1506:
1483:
1456:
1437:
1421:
1419:Bloch's Theorem
1399:
1384:
1374:
1363:
1356:
1342:
1331:
1320:
1313:
1306:
1297:)| / 24 = 1/24.
1296:
1288:
1273:
1262:
1240:
1229:
1218:
1211:
1204:
1195:)| / 24 = 1/24.
1194:
1182:
1171:
1148:
1122:
1099:
1084:
1073:
1054:
1039:
1024:
1007:, the range of
893:
880:
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827:
818:
788:
771:
693:
692:
571:
556:
533:
527:
478:
443:
430:
412:
411:
399:
386:
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376:
351:
341:
340:
307:
302:
277:)) =
249:Georges Valiron
245:
214:
196:
181:
130:
120:
119:
105:
91:defined on the
85:Bloch's theorem
69:
53:
46:
42:
35:
32:Bloch's theorem
28:
23:
22:
15:
12:
11:
5:
2133:
2131:
2123:
2122:
2117:
2107:
2106:
2103:
2102:
2083:
2062:
2061:External links
2059:
2058:
2057:
2040:(1): 608–634,
2029:
2011:(1): 275–291.
1998:
1962:
1953:
1935:(1): 671–673.
1913:
1910:
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1908:
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1772:
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1741:
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1735:
1706:Gamma function
1702:
1701:
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1678:
1673:
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1665:
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1578:
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1571:
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1564:
1561:
1556:
1552:
1546:
1543:
1512:is called the
1505:
1502:
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1454:
1435:
1420:
1417:
1394:
1382:
1369:
1361:
1354:
1345:
1344:
1340:
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1289:)| / 48 > |
1286:
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1238:
1227:
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1196:
1192:
1180:
1169:
1146:
1142:(0)| = 1, and
1120:
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1000:
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738:
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723:
720:
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714:
711:
708:
705:
701:
650:≤ 1 such that
617:
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613:
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601:
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588:
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575:
570:
567:
564:
560:
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531:
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395:
392:
389:
385:
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375:
371:
367:
364:
361:
357:
354:
349:
317:(0) = 1, and |
306:
303:
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212:
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101:
79:, a branch of
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70:
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13:
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3:
2:
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2039:
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2014:
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1991:
1987:
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1756:
1742:
1739:
1736:
1733:
1726:
1725:
1724:
1722:
1718:
1713:
1712:and Grunsky.
1711:
1707:
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1426:
1416:
1414:
1410:
1407:(0, 1/2), so
1406:
1401:
1397:
1393:
1389:
1385:
1378:
1372:
1368:
1364:
1358:) such that |
1357:
1350:
1339:
1335:
1328:
1324:
1317:
1310:
1303:
1299:
1292:
1285:
1281:
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727:
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706:
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690:
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688:
687:
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680:| = 1/3 and |
679:
674:
672:
668:
665:
661:
657:
653:
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645:
641:
636:
634:
630:
627:, and 0 <
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268:
264:
260:
256:
252:
250:
240:
238:
237:Edmund Landau
233:
231:
227:
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174:
173:biholomorphic
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