View source for Bloch constant

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

<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>

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====
<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>



====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>