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We construct regular black holes and stars that are geodesically complete and
satisfy the dominant energy condition from Einstein-$f(F^2)$ gravities with
several classes of analytic $f(F^2)$ functions that can be viewed as
perturbations to Maxwell's theory in weak field limit. We establish that
regular black holes with special static metric ($g_{tt} g_{rr}=-1$) violate the
strong energy condition and such a regular black hole with Minkowski core
violates the null energy condition. We develop a formalism to perform
electromagnetic duality transformations in $f(F^2)$. We obtain a new explicit
example where the duality is a symmetry. We study the properties of the
corresponding dyonic black hole. We study the geodesic motions of a particular
class of solutions that we call repulson stars or black holes.
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