View source for Bloch constant

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
$$
B = \inf \{ B_f : f \in H \} > 0 \ .
$$

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 [[#References|[2]]], who conjectured that the upper bound is the true value.  More recently Bonk  [[#References|[b1]]] improved the lower bound to $\frac{\sqrt{3}}{4} + 10^{-14}$.

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====
<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>
<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">[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>

</table>

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