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

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 ($g_{tt} g_{rr}=-1$) violates 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 transformation in $f(F^2)$ and obtain two new explicit examples where the duality is a symmetry. We construct the exact solution of dyonic black hole for one of them. We study the geodesic motions of a particular class of solutions that we call repulson stars or black holes.