|
|
| Line 1: |
Line 1: |
| In [[complex analysis]], a field within [[mathematics]], '''Bloch's theorem''' gives a lower bound on the size of a disc in which an inverse to a [[holomorphic function]] exists. It is named after [[André Bloch (mathematician)|André Bloch]].
| | Binder and Finisher King from Acton Vale, likes to spend time pyrotechnics, como [http://ganhedinheiro.comoganhardinheiro101.com ganhar dinheiro] na internet and swimming. Finds immense encouragement from life by touring locales like Historic Centre of Camagüey. |
| | |
| ==Statement==
| |
| Let ''f'' be a [[holomorphic function]] in the [[unit disk]] |''z''| ≤ 1. Suppose that |''f′''(0)| = 1. Then there exists a disc of radius ''b'' and an analytic function φ in this disc, such that ''f''(φ(''z'')) = ''z'' for all ''z'' in this disc. Here ''b'' > 1/72 is an absolute constant.
| |
| | |
| ==Landau's theorem==
| |
| If ''f'' is a holomorphic function in the unit disc with the property |''f′''(0)| = 1, then the image of ''f'' contains a disc of radius ''l'', where ''l'' ≥ ''b'' is an absolute constant.
| |
| | |
| This theorem is named after [[Edmund Landau]].
| |
| | |
| ==Valiron's theorem==
| |
| Bloch's theorem was inspired by the following theorem of [[Georges Valiron]]:
| |
| | |
| '''Theorem.''' If ''f'' is a non-constant entire function then there exist discs ''D'' of arbitrarily large radius and analytic functions φ in ''D'' such that ''f''(φ(''z'')) = ''z'' for ''z'' in ''D''.
| |
| | |
| Bloch's theorem corresponds to Valiron's theorem via the so-called [[Bloch's Principle]].
| |
| | |
| == Bloch's and Landau's constants ==
| |
| The lower bound 1/72 in Bloch's theorem is not the best possible. The number ''B'' defined as the [[supremum]] of all ''b'' for which this theorem holds, is called the '''Bloch's constant'''. Bloch's theorem tells us ''B'' ≥ 1/72, but the exact value of ''B'' is still unknown.
| |
| | |
| The similarly defined optimal constant ''L'' in Landau's theorem is called the '''Landau's constant'''. Its exact value is also unknown.
| |
| | |
| The best known bounds for ''B'' at present are
| |
| :<math>\frac{\sqrt{3}}{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2}} \cdot \frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})},</math>
| |
| where Γ is the [[Gamma function]]. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to [[Lars Ahlfors|Ahlfors]] and Grunsky. They also gave an upper bound for the Landau constant.
| |
| | |
| In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of ''B'' and ''L''.
| |
| | |
| == References ==
| |
| * {{cite journal|first=Lars Valerian|last=Ahlfors|authorlink=Lars Ahlfors|last2=Grunsky|first2=Helmut|author2-link=Helmut Grunsky|title=Über die Blochsche Konstante|journal=[[Mathematische Zeitschrift]]|year=1937|volume=42|number=1|pages=671–673|doi=10.1007/BF01160101}}
| |
| * {{cite conference|first=Albert II|last=Baernstein|coauthors=Vinson, Jade P.|title=Local minimality results related to the Bloch and Landau constants|booktitle=Quasiconformal mappings and analysis|place=Ann Arbor|publisher=Springer, New York|year=1998|pages=55–89}}
| |
| * {{cite journal|first=André|last=Bloch|title=Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation|journal=Annales de la faculté des sciences de l'Université de Toulouse|number=3|volume=17|year=1925|pages=1–22|issn=0240-2963}}
| |
| * {{ cite journal | first=Huaihui | last=Chen | coauthors=Gauthier, Paul M. | title=On Bloch's constant|journal=Journal d'Analyse Mathématique|year=1996|volume=69|number=1|pages=275–291|doi=10.1007/BF02787110}}
| |
| | |
| [[Category:Unsolved problems in mathematics]]
| |
| [[Category:Theorems in complex analysis]]
| |
Binder and Finisher King from Acton Vale, likes to spend time pyrotechnics, como ganhar dinheiro na internet and swimming. Finds immense encouragement from life by touring locales like Historic Centre of Camagüey.