Landau’s constant

Imagine a map that bends a round input region and may send different points to the same output. If its local stretching at the center is fixed to one, how large a round disk must appear somewhere in the image? Landau asks for the best guarantee over all such holomorphic maps of the unit disk. The covered disk need not have a one-to-one inverse patch.[1] [2]

A map can have unit local stretch at the origin and still overlap globally. What round output region is unavoidable for every such map? Let $D=\{z:|z|<1\}$ and, for holomorphic $f$ with $f^{\prime}(0)=1$, set $\rho(f)=\sup\{r:D(a,r)\subset f(D)\text{ for some }a\}$. Then $L=\inf_f\rho(f)$. The candidate is $L_+=\Gamma(1/3)\Gamma(5/6)/\Gamma(1/6)\approx0.5432589653$; the center is unrestricted and an inverse branch is unnecessary.[1] [2]

The exact guaranteed radius is unknown. A carefully chosen example gives an upper candidate about 0.543259; universal lower estimates do not reach it. Computing that candidate to more digits does not close the inequality.[1] [2]

The missing assertion is the universal lower bound L≥L₊. Faulhuber–Gumber–Shafkulovska prove ag₃(∛2,1)=2 L₊, an identity for the known candidate, not equality with the unknown infimum. Ali–Azim’s meromorphic pole-class results concern another admissible-function family and do not alter the classical target.[1] [2]

A derivative describes behavior near one point, yet here it forces coverage of an entire round region. Every point in that output disk must be attainable, even if the map folds elsewhere or gives a point several preimages. Fixing the local scale matters: without it, simply shrinking the identity map would destroy any common positive-radius guarantee.[1] [2]

The local normalization fixes a scale but imposes no global injectivity. For example, $f(z)=z+z^2$ has $f^{\prime}(0)=1$ yet sends $-1/4$ and $-3/4$ to the same value. Landau still asks how large a disk is covered, counting existence of preimages rather than selecting a unique inverse. Its extremal value measures the minimum geometric reach compatible with local analytic regularity; the cubic-mean formula relates the conjectured value to a special-function identity without proving that universal reach.[1] [2]