Bloch's theorem

The set of Bloch is a statement of the theory of functions , which in 1925 by the French mathematician André Bloch was proved. The theorem gives a limit for the complexity of the image domain of holomorphic functions .

motivation

It is an area . Then a non-constant holomorphic function is an open mapping, which means that for each pixel there is a circular disk that lies in the image . The set of Bloch exacerbated this statement to the effect that (up to normalization) independent of the function is a circular disk of a certain size in the picture area. ${\displaystyle G\subseteq \mathbb {C} }$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle f(G)}$

statement

If the unit disk and is a holomorphic function with , then the image area contains a circular disk of radius ${\displaystyle \mathbb {E} =\left\{z\in \mathbb {C} \mid |z|<1\right\}}$ ${\displaystyle f\colon {\overline {\mathbb {E} }}\rightarrow \mathbb {C} }$ ${\displaystyle f^{\prime }(0)=1}$ ${\displaystyle f(\mathbb {E} )}$ ${\displaystyle r>{\tfrac {1}{12}}.}$

Consequences

• Let it be an area and holomorphic with for one . Then contains a circular disk of radius with ${\displaystyle G\subseteq \mathbb {C} }$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle f^{\prime }(c)\neq 0}$ ${\displaystyle c\in G}$ ${\displaystyle f(G)}$ ${\displaystyle {\tfrac {1}{12}}\,\rho \,f^{\prime }(c)}$ ${\displaystyle \rho <\operatorname {dist} (c,\partial G).}$
• A non-constant whole (entirely holomorphic) function contains circular disks of arbitrarily large radii. The centers of the circles are different depending on the radius, so it is not always completely covered, for example is ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \exp(\mathbb {C} )=\mathbb {C} \setminus \{0\}.}$
• The Little Picard theorem can be proved with the help of the set of Bloch, if you do not want to rely on the results of uniformization.

Landausche constant

Bloch's theorem gives a lower bound for the radius . The question arises as to the optimal constant, i.e. which is the largest circular disk that can be accommodated in each case. For this purpose , we define for the supremum of all possible radii of circular disks that find space: ${\displaystyle r}$ ${\displaystyle f\colon {\overline {\mathbb {E} }}\rightarrow \mathbb {C} }$ ${\displaystyle f(\mathbb {E} )}$

${\displaystyle \ell (f)=\sup \left\{r\in \mathbb {R} ^{+}\mid \exists \,w\in f(\mathbb {E} ){\text{ so, dass }}B(w,r)\subseteq f(\mathbb {E} )\right\}.}$

The national constant is then defined as ${\displaystyle {\mathfrak {L}}}$

${\displaystyle {\mathfrak {L}}=\inf \left\{\ell (f)\mid f\colon {\overline {\mathbb {E} }}\rightarrow \mathbb {C} {\text{ holomorph mit }}f^{\prime }(0)=1\right\}.}$

The exact size of the constant is not known, but the following estimates can be made:

${\displaystyle {\frac {1}{2}}+10^{-335}<{\mathfrak {L}}\leq {\frac {\Gamma (1/3)\cdot \Gamma (5/6)}{\Gamma (1/6)}}=0{,}54325\ 89653\ 42976\ 70695\ \dots }$     (Follow A081760 in OEIS ),

where denotes Euler's gamma function . ${\displaystyle \Gamma }$

The upper limit was found by Raphael Robinson in 1937 (unpublished) and Hans Rademacher in 1942, who also assumed that the upper limit corresponds to the actual value of the Landauze constant. This assumption remains an open problem to this day.

Bloch's constant

According to the theorem about implicit functions , the condition in Bloch's theorem implies that an unspecified area is even mapped biholomorphically onto its image. It is therefore obvious to investigate the same question with the additional condition that the circular disk in the image area must be a biholomorphic image of an area. ${\displaystyle f^{\prime }(0)=1}$

Bloch himself achieved the estimate ${\displaystyle r>{\tfrac {1}{72}}.}$

Let us define for the supremum of all possible radii of circular disks in , which are biholomorphic images of a sub-area of : ${\displaystyle f\colon {\overline {\mathbb {E} }}\rightarrow \mathbb {C} }$ ${\displaystyle f(\mathbb {E} )}$ ${\displaystyle \mathbb {E} }$

${\displaystyle b(f)=\sup \left\{r\in \mathbb {R} ^{+}\mid \exists \,S\subseteq \mathbb {E} ,w\in f(\mathbb {E} )\colon f(S)=B(w,r){\text{ und }}f|_{S}{\text{ biholomorph}}\right\}.}$

The Bloch's constant is then defined as ${\displaystyle {\mathfrak {B}}}$

${\displaystyle {\mathfrak {B}}=\inf \left\{b(f)\mid f\colon {\overline {\mathbb {E} }}\rightarrow \mathbb {C} {\text{ holomorph mit }}f^{\prime }(0)=1\right\}.}$

The exact value of Bloch's constant is also not known, so far the estimates have been found

${\displaystyle {\frac {\sqrt {3}}{4}}+2\cdot 10^{-4}<{\mathfrak {B}}\leq {\frac {\Gamma (1/3)\cdot \Gamma (11/12)}{\Gamma (1/4)\cdot {\sqrt {1+{\sqrt {3}}}}}}=0{,}47186\ 16534\ 52681\ 78487\ \dots }$     (Follow A085508 in OEIS ).

LV Ahlfors and H. Grunsky found the upper limit in 1937. They also assumed that this limit corresponds to the actual value of Bloch's constant. This assumption has not yet been proven either.

literature

• André Bloch : Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation. Annales de la faculté des sciences de l'université de Toulouse 3 ^e série 17, 1925, pp. 1–22 (in Numdam: [1] )
• Edmund Landau : About the Bloch constant and two related global constants (22 March 1929), Mathematical Journal 30, December 1929, pp 608-634 ( " " on p 611, " " on page 614; the GDZ: [2 ] ) ${\displaystyle {\mathfrak {L}}\;<\;{\tfrac {9}{16}}}$ ${\displaystyle {\mathfrak {L}}\;\geq \;0{,}43}$
• Lars V. Ahlfors , Helmut Grunsky : About the Bloch constant (December 9, 1936), Mathematische Zeitschrift 42, December 1937, pp. 671–673 (at the GDZ: [3] )
• Lars V. Ahlfors: An extension of Schwarz's lemma (April 1, 1937), Transactions of the AMS 43, May 1938, pp. 359–364 (English; " B ≥3 ^1/2 / 4" and " L ≥1 / 2 "on p. 364; at AMS: [4] )
• Hans Rademacher : On the Bloch-Landau constant (March 21, 1942), American Journal of Mathematics 65, July 1943, pp. 387–390 (English; at Google Books: [5] )
• Albert Baernstein II , Jade P. Vinson: Local minimality results related to the Bloch and Landau constants , in Peter Duren, Juha Heinonen , Brad Osgood, Bruce Palka (Eds.): Quasiconformal mappings and analysis. A collection of papers honoring FW Gehring , Springer, New York 1998, ISBN 0-387-98299-X , pp. 55–89 (English; on Google Books: [6] )
• Steven R. Finch: Mathematical Constants. Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , p. 456
• Reinhold Remmert , Georg Schumacher: Function Theory 2 . Springer, 2006, ISBN 3-540-40432-5

Web links

• A Bloch Constant for Hyperholomorphic Functions by Dominic Rochon, June 2000 - detailed mathematical text about Bloch's constant
• Eric W. Weisstein: Landau Constant and Bloch Constant . In: MathWorld . (English)
• Sequence A159671 in OEIS ( continued fraction expansion of the Landau constant)