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An absolute constant, the existence of which is established by Bloch's theorem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166601.png" /> be the class of all holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166602.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166603.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166604.png" />. The Riemann surface of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166605.png" /> contains on one of its sheets a largest open disc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166606.png" />. It was shown by A. Bloch [[#References|[1]]] that
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An absolute constant, the existence of which is established by Bloch's theorem. Let $H$ be the class of all holomorphic functions $f(z)$ in the disc $|z| < 1$ such that $f'(0) = 1$. The Riemann surface of the function $f(z)$ contains on one of its sheets a largest open disc of radius $B_f > 0$. It was shown by A. Bloch [[#References|[1]]] that
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$$
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B = \inf \{ B_f : f \in H \} > 0 \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166607.png" /></td> </tr></table>
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The most precise known estimate for $B$ is $\frac{\sqrt{3}}{4} \le B \le 0.472$ [[#References|[2]]]. It follows from Bloch's theorem that the Riemann surface of an entire function contains single-sheeted discs of arbitrary radius; this is equivalent to the [[Picard theorem]]: for the connection between the theorems of Bloch and Picard, see e.g. [[#References|[a1]]].
 
 
The most precise known estimate is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016660/b0166608.png" /> [[#References|[2]]]. It follows from Bloch's theorem that the Riemann surface of an entire function contains single-sheeted discs of arbitrary radius; this is equivalent to the [[Picard theorem|Picard theorem]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Bloch,  "Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation"  ''Ann. Fac. Sci. Univ. Toulouse (3)'' , '''17'''  (1925)  pp. 1–22</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Ahlfors,  H. Grunsky,  "Ueber die Blochsche Konstante"  ''Math. Z.'' , '''42'''  (1937)  pp. 671–673</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Bloch,  "Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation"  ''Ann. Fac. Sci. Univ. Toulouse (3)'' , '''17'''  (1925)  pp. 1–22</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Ahlfors,  H. Grunsky,  "Ueber die Blochsche Konstante"  ''Math. Z.'' , '''42'''  (1937)  pp. 671–673</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Heins,  "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart &amp; Winston  (1962)</TD></TR>
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</table>
  
 
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====Comments====
 
For the connection between the theorems of Bloch and Picard, see e.g. [[#References|[a1]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Heins,  "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart &amp; Winston  (1962)</TD></TR></table>
 

Revision as of 17:21, 13 November 2016

An absolute constant, the existence of which is established by Bloch's theorem. Let $H$ be the class of all holomorphic functions $f(z)$ in the disc $|z| < 1$ such that $f'(0) = 1$. The Riemann surface of the function $f(z)$ contains on one of its sheets a largest open disc of radius $B_f > 0$. It was shown by A. Bloch [1] that $$ B = \inf \{ B_f : f \in H \} > 0 \ . $$

The most precise known estimate for $B$ is $\frac{\sqrt{3}}{4} \le B \le 0.472$ [2]. It follows from Bloch's theorem that the Riemann surface of an entire function contains single-sheeted discs of arbitrary radius; this is equivalent to the Picard theorem: for the connection between the theorems of Bloch and Picard, see e.g. [a1].

References

[1] A. Bloch, "Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation" Ann. Fac. Sci. Univ. Toulouse (3) , 17 (1925) pp. 1–22
[2] L.V. Ahlfors, H. Grunsky, "Ueber die Blochsche Konstante" Math. Z. , 42 (1937) pp. 671–673
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[a1] M. Heins, "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart & Winston (1962)
How to Cite This Entry:
Bloch constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bloch_constant&oldid=39739
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article