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Difference between revisions of "Bloch constant"

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(more recent value, cite Bonk (1990))
 
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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]]].
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A precise estimate for $B$ is  
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$$
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\frac{\sqrt{3}}{4} \le B \le \frac{\Gamma(1/3)\Gamma(11/12)}{\sqrt{1+\sqrt3} \Gamma(1/4)} \approx 0.4719
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$$
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due to Ahlfors and Grunsky [[#References|[2]]], who conjectured that the upper bound is the true value.  More recently Bonk  [[#References|[b1]]] improved the lower bound to $\frac{\sqrt{3}}{4} + 10^{-14}$.
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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]]].
  
 
====References====
 
====References====
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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>
 
<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>
 
<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>
 
<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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<TR><TD valign="top">[b1]</TD> <TD valign="top"> M. Bonk,  ''On Bloch’s constant'' Proc. Am. Math. Soc. '''110''' (1990) 889-894 {{DOI|10.2307/2047734}} {{ZBL|0713.30033}}</TD></TR>
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Latest revision as of 17:31, 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 $$ B = \inf \{ B_f : f \in H \} > 0 \ . $$

A precise estimate for $B$ is $$ \frac{\sqrt{3}}{4} \le B \le \frac{\Gamma(1/3)\Gamma(11/12)}{\sqrt{1+\sqrt3} \Gamma(1/4)} \approx 0.4719 $$ due to Ahlfors and Grunsky [2], who conjectured that the upper bound is the true value. More recently Bonk [b1] improved the lower bound to $\frac{\sqrt{3}}{4} + 10^{-14}$.

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)
[b1] M. Bonk, On Bloch’s constant Proc. Am. Math. Soc. 110 (1990) 889-894 DOI 10.2307/2047734 Zbl 0713.30033
How to Cite This Entry:
Bloch constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bloch_constant&oldid=39740
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article