The Geometric Function Atlas: Using What We Know About Radius Problems to Solve New Ones

Abstract

Usually, radius problems between Ma--Minda classes are solved one pair at a time, by fitting a disk inside the target region, providing a sufficient but not always the sharp radius. The same class appearing under different names makes comparison of results harder, with mistakes and duplication eluding detection. The Geometric Function Atlas catalogs radius problems using exact generators and a direction (source $\to$ target), ensuring unique identification. For each problem, we then compute the radius numerically, together with the point where the image first meets the target boundary. This helps us identify the sharp estimate which guides the proof we write. With this system, we were able to refine two earlier stated estimates, the disk radius for the sine $\to$ modified sigmoid and sine $\to$ rational cases from $\mathrm{arsinh}c$ to $\arcsin c$, and correct earlier stated values for the crescent and exponential classes: the sharp radius is $\sin 1$ from crescent to exponential, and $\log (1+\sqrt{2})$ from exponential to crescent. All four proofs follow a common strategy: an estimate that is exact at the contact point, followed by the standard extremal function. Applying the method, we derive fifteen further sharp radii, four of them known.