965:
777:
1185:
1426:
427:
1737:
1929:
618:
1818:
1618:
1262:
309:
1542:
555:
1578:
1660:
347:
1509:
746:
522:
136:
1291:
644:
960:{\displaystyle L_{z}(\varphi ,v):=\sum _{k,l=1}^{n}{\frac {\partial ^{2}\varphi }{\partial z_{k}\partial {\overline {z}}_{l}}}(z)v_{k}{\overline {v}}_{l}\ \ (v\in C^{n}),}
772:
270:
1453:
196:
1053:
985:
710:
667:
687:
575:
160:
228:
2085:
1845:
1317:
1476:
1008:
493:
450:
1865:
1030:
470:
1058:
106:
In the more recent times several general theorems were proved which can be regarded as rigorous statements in the spirit of the Bloch
Principle:
81:
77:
2118:
1322:
2113:
355:
1664:
53:
occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in
39:
1870:
580:
1742:
2056:
Zalcman's lemma in Cn, Complex
Variables and Elliptic Equations, 65:5, 796-800, DOI: 10.1080/17476933.2019.1627529
1946:
1583:
1190:
277:
1514:
527:
61:
1481:
1547:
73:
87:
Based on his
Principle, Bloch was able to predict or conjecture several important results such as the
1962:
1622:
1484:, which states that a family is normal if and only if the spherical derivatives are locally bounded:
316:
1490:
715:
503:
117:
1267:
623:
69:
751:
232:
2079:
2067:
1431:
169:
1970:
1993:
Bloch, A. (1926). "La conception actuelle de la theorie de fonctions entieres et meromorphes".
1035:
970:
695:
649:
497:
Zalcman's lemma may be generalized to several complex variables. First, define the following:
100:
672:
560:
145:
2059:
2036:
203:
65:
1823:
2013:
1950:
50:
1296:
1458:
990:
475:
432:
1850:
1015:
455:
88:
17:
2107:
2071:
1954:
1943:
96:
2040:
92:
2063:
1180:{\displaystyle f^{\sharp }(z):=\sup _{|v|=1}{\sqrt {L_{z}(\log(1+|f|^{2}),v)}}.}
139:
35:
31:
49:
and explains this as follows: Every proposition in whose statement the
2027:
Zalcman, L. (1975). "Heuristic principle in complex function theory".
620:
contains either a subsequence which converges to a limit function
1480:
The following characterization of normality can be made based on
1421:{\displaystyle f^{\sharp }(z):={\frac {|f'(z)|}{1+|f(z)|^{2}}}}
99:'s result that an exceptional set of radii is unavoidable in
1969:
such that every holomorphic map from the unit disc with the
1602:
1496:
605:
509:
296:
123:
422:{\displaystyle f_{n}(z_{n}+\rho _{n}\zeta )\to g(\zeta ),}
1732:{\displaystyle \rho _{j}=1/f_{j}^{\sharp }(z_{j})\to 0,}
95:'s theorem on holomorphic curves omitting hyperplanes,
47:
Nihil est in infinito quod non prius fuerit in finito,
1873:
1853:
1826:
1745:
1667:
1625:
1586:
1550:
1517:
1493:
1461:
1434:
1325:
1299:
1270:
1193:
1061:
1038:
1018:
993:
973:
780:
754:
718:
698:
675:
652:
626:
583:
563:
530:
506:
478:
458:
435:
358:
319:
280:
235:
206:
172:
148:
120:
60:
Bloch mainly applied this principle to the theory of
1924:{\displaystyle g^{\sharp }(z)\leq g^{\sharp }(0)=1}
68:. Thus, for example, according to this principle,
1923:
1859:
1839:
1812:
1731:
1654:
1612:
1572:
1536:
1503:
1470:
1447:
1420:
1311:
1285:
1256:
1187:This quantity is well defined since the Levi form
1179:
1047:
1024:
1002:
979:
959:
766:
740:
704:
681:
661:
638:
612:
569:
549:
516:
487:
464:
444:
421:
341:
303:
264:
222:
190:
154:
130:
613:{\displaystyle \{f_{j}\}\subseteq {\mathcal {F}}}
1813:{\displaystyle g_{j}(z)=f_{j}(z_{j}+\rho _{j}z)}
1085:
669:or a subsequence which converges uniformly to
8:
597:
584:
429:spherically uniformly on compact subsets of
2084:: CS1 maint: numeric names: authors list (
967:and call it the Levi form of the function
162:is not normal if and only if there exist:
2010:Introduction to complex hyperbolic spaces
1900:
1878:
1872:
1852:
1831:
1825:
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1785:
1772:
1750:
1744:
1711:
1698:
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1601:
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1591:
1585:
1555:
1549:
1528:
1516:
1495:
1494:
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1460:
1439:
1433:
1409:
1404:
1386:
1373:
1351:
1348:
1330:
1324:
1298:
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1236:
1231:
1222:
1198:
1192:
1154:
1149:
1140:
1116:
1110:
1097:
1089:
1088:
1066:
1060:
1037:
1017:
992:
972:
945:
920:
910:
903:
881:
871:
861:
843:
836:
830:
813:
785:
779:
753:
723:
717:
697:
674:
651:
625:
604:
603:
591:
582:
562:
541:
529:
508:
507:
505:
477:
472:is a nonconstant meromorphic function on
457:
434:
389:
376:
363:
357:
324:
318:
295:
294:
285:
279:
251:
245:
236:
234:
211:
205:
171:
147:
122:
121:
119:
1613:{\displaystyle f_{j}\in {\mathcal {F}},}
1257:{\displaystyle L_{z}(\log(1+|f|^{2}),v)}
45:Bloch states the principle in Latin as:
1985:
1455:coincides with the spherical metric on
304:{\displaystyle f_{n}\in {\mathcal {F}}}
2077:
524:of holomorphic functions on a domain
7:
1537:{\displaystyle \Omega \subset C^{n}}
646:uniformly on each compact subset of
550:{\displaystyle \Omega \subset C^{n}}
1901:
1879:
1847:to a non-constant entire function
1699:
1564:
1518:
1440:
1331:
1277:
1067:
1039:
867:
854:
840:
761:
732:
676:
653:
633:
564:
531:
149:
25:
1961:is constant. Then there exists a
1573:{\displaystyle z_{0}\in \Omega .}
1319:the above formula takes the form
1997:. Vol. 25. pp. 83–103.
1820:converges locally uniformly in
1655:{\displaystyle z_{j}\to z_{0},}
577:if every sequence of functions
342:{\displaystyle \rho _{n}\to 0+}
2041:10.1080/00029890.1975.11993942
1912:
1906:
1890:
1884:
1807:
1778:
1762:
1756:
1720:
1717:
1704:
1636:
1504:{\displaystyle {\mathcal {F}}}
1405:
1400:
1394:
1387:
1374:
1370:
1364:
1352:
1342:
1336:
1251:
1242:
1232:
1223:
1213:
1204:
1169:
1160:
1150:
1141:
1131:
1122:
1098:
1090:
1078:
1072:
951:
932:
896:
890:
803:
791:
741:{\displaystyle C^{2}(\Omega )}
735:
729:
517:{\displaystyle {\mathcal {F}}}
413:
407:
401:
398:
369:
330:
252:
237:
131:{\displaystyle {\mathcal {F}}}
89:Ahlfors's Five Islands theorem
1:
2064:10.1080/17476933.2019.1627529
1977:does not increase distances.
1286:{\displaystyle z\in \Omega .}
639:{\displaystyle f\neq \infty }
1544:is not normal at some point
1511:of functions holomorphic on
915:
876:
767:{\displaystyle z\in \Omega }
265:{\displaystyle |z_{n}|<r}
1580:Then there exist sequences
1448:{\displaystyle z^{\sharp }}
191:{\displaystyle 0<r<1}
2135:
2119:Philosophy of mathematics
1947:complex analytic manifold
1487:Suppose that the family
1048:{\displaystyle \Omega ,}
980:{\displaystyle \varphi }
705:{\displaystyle \varphi }
689:on each compact subset.
662:{\displaystyle \Omega ,}
2114:Mathematical principles
1739:such that the sequence
1264:is nonnegative for all
682:{\displaystyle \infty }
570:{\displaystyle \Omega }
155:{\displaystyle \Delta }
2054:P. V. Dovbush (2020).
1925:
1861:
1841:
1814:
1733:
1656:
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1538:
1505:
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1449:
1422:
1313:
1287:
1258:
1181:
1049:
1026:
1004:
981:
961:
835:
768:
742:
706:
683:
663:
640:
614:
571:
551:
518:
489:
466:
446:
423:
343:
305:
266:
224:
223:{\displaystyle z_{n},}
192:
156:
132:
18:Bloch's Principle
1933:
1926:
1862:
1842:
1840:{\displaystyle C^{n}}
1815:
1734:
1657:
1615:
1575:
1539:
1506:
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1450:
1423:
1314:
1288:
1259:
1182:
1050:
1027:
1005:
982:
962:
809:
769:
748:define at each point
743:
707:
684:
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615:
572:
552:
519:
490:
467:
447:
424:
344:
306:
267:
225:
193:
157:
133:
109:
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1584:
1548:
1515:
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1323:
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1191:
1059:
1036:
1016:
991:
971:
778:
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581:
561:
528:
504:
476:
456:
433:
356:
317:
278:
233:
204:
170:
146:
118:
2029:Amer. Math. Monthly
1703:
1312:{\displaystyle n=1}
1293:In particular, for
692:For every function
1949:, such that every
1921:
1857:
1837:
1810:
1729:
1689:
1652:
1610:
1570:
1534:
1501:
1471:{\displaystyle C.}
1468:
1445:
1418:
1309:
1283:
1254:
1177:
1109:
1045:
1032:is holomorphic on
1022:
1003:{\displaystyle z.}
1000:
977:
957:
764:
738:
702:
679:
659:
636:
610:
567:
547:
514:
488:{\displaystyle C.}
485:
462:
445:{\displaystyle C,}
442:
419:
339:
301:
262:
220:
188:
152:
128:
74:Schottky's theorem
2008:Lang, S. (1987).
1995:Enseignement Math
1860:{\displaystyle g}
1416:
1172:
1084:
1025:{\displaystyle f}
931:
928:
918:
888:
879:
774:a Hermitian form
465:{\displaystyle g}
142:on the unit disc
101:Nevanlinna theory
78:Valiron's theorem
28:Bloch's Principle
16:(Redirected from
2126:
2099:
2096:
2090:
2089:
2083:
2075:
2051:
2045:
2044:
2024:
2018:
2017:
2005:
1999:
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1930:
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1922:
1905:
1904:
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1838:
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1802:
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1716:
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1647:
1635:
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1619:
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1606:
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1510:
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1502:
1500:
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1474:
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1451:
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1444:
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1427:
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1419:
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1408:
1390:
1378:
1377:
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1335:
1334:
1318:
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1310:
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1202:
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1173:
1159:
1158:
1153:
1144:
1121:
1120:
1111:
1108:
1101:
1093:
1071:
1070:
1054:
1052:
1051:
1046:
1031:
1029:
1028:
1023:
1009:
1007:
1006:
1001:
986:
984:
983:
978:
966:
964:
963:
958:
950:
949:
929:
926:
925:
924:
919:
911:
908:
907:
889:
887:
886:
885:
880:
872:
866:
865:
852:
848:
847:
837:
834:
829:
790:
789:
773:
771:
770:
765:
747:
745:
744:
739:
728:
727:
711:
709:
708:
703:
688:
686:
685:
680:
668:
666:
665:
660:
645:
643:
642:
637:
619:
617:
616:
611:
609:
608:
596:
595:
576:
574:
573:
568:
556:
554:
553:
548:
546:
545:
523:
521:
520:
515:
513:
512:
494:
492:
491:
486:
471:
469:
468:
463:
451:
449:
448:
443:
428:
426:
425:
420:
394:
393:
381:
380:
368:
367:
348:
346:
345:
340:
329:
328:
310:
308:
307:
302:
300:
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290:
289:
271:
269:
268:
263:
255:
250:
249:
240:
229:
227:
226:
221:
216:
215:
197:
195:
194:
189:
161:
159:
158:
153:
137:
135:
134:
129:
127:
126:
70:Picard's theorem
66:complex variable
21:
2134:
2133:
2129:
2128:
2127:
2125:
2124:
2123:
2104:
2103:
2102:
2097:
2093:
2076:
2053:
2052:
2048:
2026:
2025:
2021:
2014:Springer Verlag
2007:
2006:
2002:
1992:
1991:
1987:
1983:
1971:Poincaré metric
1951:holomorphic map
1936:
1896:
1874:
1869:
1868:
1849:
1848:
1827:
1822:
1821:
1794:
1781:
1768:
1746:
1741:
1740:
1707:
1668:
1663:
1662:
1639:
1626:
1621:
1620:
1587:
1582:
1581:
1551:
1546:
1545:
1524:
1513:
1512:
1489:
1488:
1482:Marty's theorem
1457:
1456:
1435:
1430:
1429:
1403:
1379:
1356:
1350:
1326:
1321:
1320:
1295:
1294:
1266:
1265:
1230:
1194:
1189:
1188:
1148:
1112:
1062:
1057:
1056:
1034:
1033:
1014:
1013:
989:
988:
969:
968:
941:
909:
899:
870:
857:
853:
839:
838:
781:
776:
775:
750:
749:
719:
714:
713:
694:
693:
671:
670:
648:
647:
622:
621:
587:
579:
578:
559:
558:
537:
526:
525:
502:
501:
474:
473:
454:
453:
431:
430:
385:
372:
359:
354:
353:
320:
315:
314:
281:
276:
275:
241:
231:
230:
207:
202:
201:
168:
167:
144:
143:
116:
115:
112:
110:Zalcman's lemma
82:Bloch's theorem
80:corresponds to
72:corresponds to
51:actual infinity
23:
22:
15:
12:
11:
5:
2132:
2130:
2122:
2121:
2116:
2106:
2105:
2101:
2100:
2091:
2046:
2035:(8): 813–817.
2019:
2000:
1984:
1982:
1979:
1935:
1932:
1920:
1917:
1914:
1911:
1908:
1903:
1899:
1895:
1892:
1889:
1886:
1881:
1877:
1856:
1834:
1830:
1809:
1806:
1801:
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1793:
1788:
1784:
1780:
1775:
1771:
1767:
1764:
1761:
1758:
1753:
1749:
1728:
1725:
1722:
1719:
1714:
1710:
1706:
1701:
1696:
1692:
1687:
1683:
1680:
1675:
1671:
1651:
1646:
1642:
1638:
1633:
1629:
1609:
1604:
1599:
1594:
1590:
1569:
1566:
1563:
1558:
1554:
1531:
1527:
1523:
1520:
1498:
1467:
1464:
1442:
1438:
1412:
1407:
1402:
1399:
1396:
1393:
1389:
1385:
1382:
1376:
1372:
1369:
1366:
1362:
1359:
1354:
1347:
1344:
1341:
1338:
1333:
1329:
1308:
1305:
1302:
1282:
1279:
1276:
1273:
1253:
1250:
1247:
1244:
1239:
1234:
1229:
1225:
1221:
1218:
1215:
1212:
1209:
1206:
1201:
1197:
1176:
1171:
1168:
1165:
1162:
1157:
1152:
1147:
1143:
1139:
1136:
1133:
1130:
1127:
1124:
1119:
1115:
1107:
1104:
1100:
1096:
1092:
1087:
1083:
1080:
1077:
1074:
1069:
1065:
1044:
1041:
1021:
999:
996:
976:
956:
953:
948:
944:
940:
937:
934:
923:
917:
914:
906:
902:
898:
895:
892:
884:
878:
875:
869:
864:
860:
856:
851:
846:
842:
833:
828:
825:
822:
819:
816:
812:
808:
805:
802:
799:
796:
793:
788:
784:
763:
760:
757:
737:
734:
731:
726:
722:
701:
678:
658:
655:
635:
632:
629:
607:
602:
599:
594:
590:
586:
566:
544:
540:
536:
533:
511:
484:
481:
461:
441:
438:
418:
415:
412:
409:
406:
403:
400:
397:
392:
388:
384:
379:
375:
371:
366:
362:
350:
349:
338:
335:
332:
327:
323:
311:
298:
293:
288:
284:
272:
261:
258:
254:
248:
244:
239:
219:
214:
210:
198:
187:
184:
181:
178:
175:
151:
125:
111:
108:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2131:
2120:
2117:
2115:
2112:
2111:
2109:
2095:
2092:
2087:
2081:
2073:
2069:
2065:
2061:
2057:
2050:
2047:
2042:
2038:
2034:
2030:
2023:
2020:
2015:
2011:
2004:
2001:
1996:
1989:
1986:
1980:
1978:
1976:
1972:
1968:
1964:
1960:
1956:
1955:complex plane
1952:
1948:
1945:
1941:
1934:Brody's lemma
1931:
1918:
1915:
1909:
1897:
1893:
1887:
1875:
1854:
1832:
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36:mathematics
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352:such that
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1953:from the
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200:points
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