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 [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