Abstract
Every nonconstant meromorphic function in the plane univalently covers spherical discs of radii arbitrarily close to arctan p 8… 70 ‐ 32 0 . If in addition all critical points of the function are multiple, then a similar statement holds with …=2. These constants are the best possible. The proof is based on the consideration of negatively curved singular surfaces associated with meromorphic functions.
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