An Improved Upper Bound for the Univalent Bloch-Landau Constant

We exhibit a disc with 24 slits, 18 of them curved, and give a computer-assisted proof that its ratio of inradius to conformal radius is less than 0.65628589. This is below the best upper bound for the univalent Bloch–Landau constant previously stated as a theorem, 0.6563937 (Carroll and Ortega-Cerdà, 2009, by numerical evaluation of an explicit formula), and below the best numerical value 0.656319277 in the literature (Ortega-Cerdà and Pridhnani, 2010). In the proof, the inradius is bounded by an exact covering argument, and the conformal radius is bounded below by a comparison function whose boundary values are controlled in ball arithmetic. The domain was found by an evolutionary search driven by a spectral integral-equation solver.