Bloch’s constant

A map may bend a round input region and send several points to the same output. If its local stretching at the center is one, how large a round output disk can always be reached from a single one-to-one input patch? Bloch asks for the best guarantee over holomorphic maps of the unit disk. The patch can lie anywhere; the whole map need not be one-to-one.[1]

Unit local stretching need not prevent overlaps elsewhere. What radius can always be recovered as a round region with a single analytic inverse branch on some input patch? Put $D=\{z:|z|<1\}$ and $\beta(f)=\sup\{r>0:f\text{ maps some }\Omega\subset D\text{ biholomorphically onto a disk of radius }r\}$. Determine $B=\inf\{\beta(f):f\text{ holomorphic on }D,\ f^{\prime}(0)=1\}$. Neither global injectivity nor a prescribed image center is required.[2]

The exact best guarantee is unknown, even though an algorithm can approximate the constant arbitrarily closely.[1] [2]

Computability and the recent interval-arithmetic lower-bound claim do not yield the exact extremal constant. Ordinary B is distinct from Landau’s L and locally univalent B∞.[3] [4]

Within the guaranteed patch, every point of the output disk has one input and an analytic inverse can recover it. Mere image coverage can leave several competing inputs. Bloch’s theorem therefore turns information about stretching at one point into a controlled region for reversing the map, even when the complete map overlaps.[2]

Local inversion only asserts a sufficiently small neighborhood for a specified map. Bloch’s constant supplies a uniform quantitative scale across the entire normalized holomorphic family. The disk has a holomorphic inverse into the chosen subdomain, even if another input elsewhere reaches the same output. Thus $B$ measures reversible coverage, whereas Landau’s $L$ measures image coverage alone. Computability offers numerical access to this universal scale; it does not replace the question of its sharp geometric value.[2]