3 Added conjectured value for Landau constant.

The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance Remmert Funktionentheorie II or Steven Finch marvelous "Mathematical Constants") was conjectured by Ahlfors to be

$$\frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$$

(This value, if I remember well Ahlfors' article corresponds to a particular function that he constructed for this purpose).

The Bloch Constant $B$ is currently known to be at least slightly greater than $\frac{\sqrt{3}}{4}$ (several articles improving upon each other by Mario Bonk, Chen and Gauthier, Xiong).

Has there been some progress since 1998 on the lower bound ?

Same question for the closely related (univalent) Landau constant (quite often called Bloch-Landau constant) , sometimes seen as $B_\infty$) ?

The conjectured upper bound is

$$\frac{\Gamma(\frac{1}{3})\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6})}$$

What can be said of the various adaptations or specializations of this constant to various class of functions, and extensions of these constants to several complex variables or other functional spaces ?

I give as background the original article from Bloch, Ahlfors and Grunsky.

(1) A. Bloch, Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation, Ann. Fac. Sci. Univ. Toulouse, vol. 17, (1925), pp1-22.

(2) L. V. Ahlfors and H. Grunsky, Über die Blochsche Konstante, Math. Zeitschrift 42 (1937), pp671–673.

(3) L. V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), pp359–364.

(these two are reprinted in Ahlfors Works vol 1)

Ahlfors life and works are evocated in an AMS Notices of 1998.

2 added original references, improved TeX.

The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance Remmert Funktionentheorie II or Steven Finch marvelous "Mathematical Constants") was conjectured by Ahlfors to be

$\frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$$\frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$$ (This value, if I remember well Ahlforsand Grunsky ' article corresponds to a particular function that he constructed for this purpose). The Block Bloch Constant is currently known to be at least slightly greater than$\frac{\sqrt{3}}{4}$(several articles improving upon each other by Mario Bonk, Chen and Gauthier, Xiong). Has there been some progress since 1998 on the lower bound ? Same question for the closely related (univalent) Landau constant (quite often called Bloch-Landau constant) ? What can be said of the various adaptations or specializations of this constant to various class of functions ? I give as background the original article from Bloch, Ahlfors and Grunsky. (1) A. Bloch, Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation, Ann. Fac. Sci. Univ. Toulouse, vol. 17, (1925), pp1-22. (2) L. V. Ahlfors and H. Grunsky, Über die Blochsche Konstante, Math. Zeitschrift 42 (1937), pp671–673. (3) L. V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), pp359–364. (these two are reprinted in Ahlfors Works vol 1) Ahlfors life and works are evocated in an AMS Notices of 1998. 1 # Current status of Bloch Constant and Landau Constant bounds The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance Remmert Funktionentheorie II or Steven Finch marvelous "Mathematical Constants") was conjectured by Ahlfors to be$\frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$(This value, if I remember well Ahlfors and Grunsky article corresponds to a particular function that he constructed for this purpose). The Block Constant is known to be at least slightly greater than$\frac{\sqrt{3}}{4}\$ (several articles improving upon each other by Mario Bonk, Chen and Gauthier, Xiong).

Has there been some progress since 1998 on the lower bound ?

Same question for the closely related Landau constant ?

What can be said of the various adaptations or specializations of this constant to various class of functions ?