  One of the long standing problems is a quest for regular black hole
solutions, in which a resolution of the spacetime singularity has been achieved
by some physically reasonable, classical field, before one resorts to the
quantum gravity. The prospect of using nonlinear electromagnetic fields for
this goal has been limited by the Bronnikov's no-go theorems, focused on
Lagrangians depending on the electromagnetic invariant $F_{ab}F^{ab}$ only. We
extend Bronnikov's results by taking into account Lagrangians that depend on
both electromagnetic invariants, $F_{ab}F^{ab}$ and $F_{ab}\,{\star F^{ab}}$,
and prove that the tension between the Lagrangian's Maxwellian weak field limit
and boundedness of the curvature invariants persists in more general class of
theories.
