Rectangular-lattice Landau extremal

Definitions and mathematical statement

For $a\in(0,1)$ put $b=\sqrt{1-a^2}>0$ and let \[ \Lambda_a=\{an+ibm:n,m\in\mathbb Z\},\qquad \mathbb D=\{z\in\mathbb C:|z|<1\}. \] Let $f_a:\mathbb D\to\mathbb C\setminus\Lambda_a$ be the holomorphic universal covering map normalized by \[ f_a(0)=\frac{a+ib}{2},\qquad f_a'(0)\in(0,\infty). \] Here $a,b$ are the positive side lengths of the rectangular lattice and $a^2+b^2=1$ fixes its scale. A universal covering map is a surjective holomorphic covering whose domain is simply connected; the stated normalization fixes the covering map.

Determine the extremal value \[ M=\sup_{0<a<1}f_a'(0) \] and the parameters $a$ at which it is attained. In particular, establish attainment if the requested maximum is to be realized inside $(0,1)$ and compute its exact value. The supplied summary describes degenerate/asymptotic cases as understood but does not supply their formulas; it leaves this interior optimization unresolved.

Short English statement

Among rectangular lattices whose squared side lengths sum to one, maximize the derivative at the center-normalized universal covering of the lattice complement. Determine the exact value and the maximizing lattice shapes.

Sources

• Alexandre Eremenko, My favorite unsolved problems, item "Rectangular lattice covering problem," https://www.math.purdue.edu/~eremenko/uns1.html; accessed 2026-07-29. The supplied provenance records PDF-text extraction from the indexed source.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $3700038$, source code catalogue entry, Rectangular-lattice Landau extremal. The record assigns proposed year 2009 and the proposer field "Alexandre Eremenko and cited proposers." Dataset attribution: Ulam AI and the UnsolvedMath Contributors.