Fran Wikström: How large a disc?

Abstract:  If \(f\) is holomorphic on the unit disc, with \(f'(0) = 1\), what can we say about the existence of large discs in the image of \(f\)? Depending on the exact formulation of this question, we are led to a family of classical ''Bloch style'' constants, which have been studied since the 1920's, but the exact values of these constants are still wide open.

I will tell you about the history of these problems and recent progress in obtaining better lower bounds for the Bloch, Landau and, in particular, the univalent Bloch-Landau constants. The latter one is really about the geometry of simply connected domains. Landau proved in 1929 that if \(f\) is univalent, the image contains a disc of radius at least \(0.5682\), and despite many attempts, the best lower bound has stood at \(0.5708858\) since 2009. All previous work has gone into finding the largest disc in the image that is centered at \(f(0)\), but by allowing translated disc, we obtain a substantial improvement, showing that the image contains discs of radius at least \(0.619\), thus closing more than half the gap to the best upper bound (\(0.6562681\)).

The full result requires computer-assisted computations, but I will sketch the ideas of a ''by hand'' argument that gives a lower bound of \(0.592\) and explain what is needed for the computational improvements.