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

This is Andre Bloch most famous theorem, which has been published in 1925 in the Annales de la Faculte des Sciences Toulouse. The optimal absolute constant b is still unknown today, despite numerous works and steady progress (see e.g. Bonk, M. ; Eremenko, A. ; Covering properties of meromorphic functions, negative curvature and spherical geometry. Ann. of Math. (2) 152 (2000), no. 2, 551-592).
Andre Bloch was an unusual mathematician. You'll nd some biographical information in this article by H. Cartan and J. Ferrand.