  We consider a two-dimensional analogue of Jacobi theta functions and prove
that, among all lattices $\Lambda \subset \mathbb{R}^2$ with fixed density, the
minimal value is maximized by the hexagonal lattice. This result can be
interpreted as the dual of a 1988 result of Montgomery who proved that the
hexagonal lattice minimizes the maximal values. Our inequality resolves a
conjecture of Strohmer and Beaver about the operator norm of a certain type of
frame in $L^2(\mathbb{R})$. It has implications for minimal energies of ionic
crystals studied by Born, the geometry of completely monotone functions and a
connection to the elusive Landau constant.
