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We construct regular black holes and stars that are geodesically complete andsatisfy the dominant energy condition from Einstein-$f(F^2)$ gravities withseveral classes of analytic $f(F^2)$ functions that can be viewed asperturbations to Maxwell's theory in weak field limit. We establish thatregular black holes with special static metric ($g_{tt} g_{rr}=-1$) violate thestrong energy condition and such a regular black hole with Minkowski coreviolates the null energy condition. We develop a formalism to performelectromagnetic duality transformations in $f(F^2)$. We obtain two new explicitexamples where the duality is a symmetry. We study the properties of thecorresponding dyonic black holes. We study the geodesic motions of a particularclass of solutions that we call repulson stars or black holes.

Regular Black Holes and Stars from Analytic $f(F^2)$