QED in strong, finite-flux magnetic fields

Lower bounds are placed on the fermionic determinants of Euclidean quantum electrodynamics in two and four dimensions in the presence of a smooth, finite-flux, static, unidirectional magnetic field $B(r) =(0,0,B(r))$, where $B(r) \geq 0$ or $B(r) \leq 0$, and $r$ is a point in the xy-plane.