## Current status of Bloch Constant and Landau Constant bounds

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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.

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I am not an expert but perhaps this paper could be of some interest: R. Rettinger, On the computability of Bloch's constant ( dx.doi.org/10.1016/j.entcs.2008.03.024) – mathphysicist May 12 2010 at 0:04
And another paper that could be of interest: B. Skinner, The univalent Bloch constant problem. Complex Var. Elliptic Equ. 54, No. 10, 951-955 (2009). DOI: dx.doi.org/10.1080/17476930903197199  Summary: Suppose f∈S and |f(z)|≥B(|z|), where B is a nonnegative function on [0, 1). We present a theorem which provides an implicit function C(|z|) such that |f(z)|≥C(|z|)≥B(|z|). We use this theorem to obtain an explicit improvement in the lower bound for the univalent Bloch constant to 0.5708858. – mathphysicist May 12 2010 at 0:18
@mathphysicist: thanks, especially for the second reference which is about a lower bound to the Bloch-Landau constant. I then found a paper: euclid.ucc.ie/pages/staff/carroll/Papers/…  by Tom Carrol and Joaquim Ortega-Cerda on an upper bound for L giving 0.6563937. They use a construction with a threefold symmetry which recalls Ahlfors construction. I will try to add more background about L in the original question.  – ogerard May 12 2010 at 5:53

## 1 Answer

The world record on the Bloch's constant seems to be MR1690898 by C. Xiong, who proved $B\geq \sqrt{3}/4+3.10^{-4}$. I recall that $\sqrt{3}/4$ is the Ahlfors estimate, then Heins proved that $B$ is strictly greater than that, and Bonk was the first to prove this with a specific constant. Then this constant was slightly improved, first by Gauthier and Chen MR1428103 and then by Xiong.

This is far from the conjectured value. There are also results which show that the Ahlfors - Grunsky conjectured extremal function gives a local extremum for certain variations. But the classes of variations considered are narrow. For example, Baernstein II and Vinson proved that the Ahlfors Grunsky function gives a local extremum for the class of ramified coverings which are ramified only over some lattice points. (So the hexagonal lattice is locally extremal for such restricted problem).

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