Generalisation for regular black holes on general relativity to f (R) gravity

In this paper, we determine regular black hole solutions using a very general f (R) theory, coupled to a non-linear electromagnetic field given by a Lagrangian LNED. The functions f (R) and LNED are in principle left unspecified. Instead, the model is constructed through a choice of the mass function M(r) presented in the metric coefficients. Solutions which have a regular behaviour of the geometric invariants are found. These solutions have two horizons, the event horizon and the Cauchy horizon. All energy conditions are satisfied in the whole space-time, except the strong energy condition (SEC), which is violated near the Cauchy horizon. We present also a new theorem related to the energy conditions in f (R) gravity, re-obtaining the well-known conditions in the context of general relativity when the geometry of the solution is the same.