A note on non-linear electrodynamics, regular black holes and the entropy function

We examine four dimensional magnetically charged extremal black holes in certain non-linear U(1) gauge theories coupled to two derivative gravity. For a given coupling, one can tune the magnetic charge (or vice versa) so that the curvature singularity at the centre of the space-time is cancelled. Since these solutions have a horizon but no singularity, they have been called regular black holes. Contrary to recent claims in the literature, we find that the entropy function formalism reproduces the near horizon geometry and gives the correct entropy for these objects.