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January 2021

Let ƒ be a holomorphic function in the unit disk of the complex plane fixing the origin with differential of modulus 1. The local inversion theorem guarantees that ƒ admits a holomorphic inverse g in a small disk of radius r > 0 centered at the origin. André Bloch showed that the maximum radius of existence of g is bounded below by an absolute constant b > 1/72.

This is André Bloch most famous theorem, which has been published in 1925 in the Annales de la Faculté des Sciences Toulouse. The optimal absolute constant b is still unknown today, despite numerous works and steadyprogress (see e.g. Bonk, M. ; Eremenko, A. ; Covering properties of meromor-phic functions, negative curvature and spherical geometry. Ann. of Math. 2000, 152 , no. 2, 551-592).

André Bloch was an unusual mathematician. You’ll find some biographical information in this article by H.Cartan and J.Ferrand.