  We consider a wide family of non-uniformly expanding maps and hyperbolic
H\"older continuous potentials. We prove that the unique equilibrium state
associated to each element of this family is given by the eigenfunction of the
transfer operator and the eigenmeasure of the dual operator (both having the
spectral radius as eigenvalue). We show that the transfer operator has the
spectral gap property in some space of H\"older continuous observables and from
this we obtain an exponential decay of correlations and a central limit theorem
for the equilibrium state. Moreover, we establish the analyticity with respect
to the potential of the equilibrium state as well as that of other
thermodynamic quantities. Furthermore, we derive similar results for the
equilibrium state associated to a family of non-uniformly hyperbolic skew
products and hyperbolic H\"older continuous potentials.
