“Schottky's theorem”的意思、由来-开放百科全书

词条

Schottky's theorem

释义

References

In mathematical complex analysis, Schottky's theorem, introduced by {{harvs|txt|authorlink=Friedrich Schottky|last=Schottky|year=1904}} is a quantitative version of Picard's theorem which states that the size |f(z)| of a holomorphic function f in the open unit disk that does not take the values 0 or 1 can be bounded in terms of z and f(0).

Schottky's original theorem did not give an explicit bound for f. {{harvs|txt|last=Ostrowski|year1=1931|year2=1933}} gave some weak explicit bounds. {{harvtxt|Ahlfors|1938|loc=theorem B}} gave a strong explicit bound, showing that if f is holomorphic in the open unit disk and does not take the values 0 or 1 then

.

Several authors, such as {{harvtxt|Jenkins|1955}}, have given variations of Ahlfors's bound with better constants: in particular {{harvtxt|Hempel|1980}} gave some bounds whose constants are in some sense the best possible.

References

{{Citation | last1=Ahlfors | first1=Lars V. | title=An Extension of Schwarz's Lemma | jstor=1990065 | year=1938 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=43 | issue=3 | pages= 359–364 | doi=10.2307/1990065}}
{{Citation | last1=Hempel | first1=Joachim A. | title=Precise bounds in the theorems of Schottky and Picard | doi=10.1112/jlms/s2-21.2.279 |mr=575385 | year=1980 | journal=Journal of the London Mathematical Society | issn=0024-6107 | volume=21 | issue=2 | pages=279–286}}
{{Citation | last1=Jenkins | first1=J. A. | title=On explicit bounds in Schottky's theorem | doi=10.4153/CJM-1955-010-4 |mr=0066460 | year=1955 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=7 | pages=76–82}}
{{Citation | last1=Ostrowski | first1=A. M. | title=Studien über den schottkyschen satz | url=https://books.google.com/books?id=uzwgAAAAIAAJ | publisher= Basel, B. Wepf & cie. | year=1931}}
{{Citation | last1=Ostrowski | first1=Alexander | title=Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte 0 und 1 nicht annimmt | doi=10.5169/seals-6655 | year=1933 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=5 | pages=55}}
{{Citation | last1=Schottky | first1=F. | title=Über den Picardschen Satz und die Borelschen Ungleichungen | year=1904 | journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin | pages=1244–1263}}

1 : Theorems in complex analysis

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。