Loop quantum cosmology of k=1 FLRW: effects of inverse volume corrections

We consider the k = 1 Friedman-Lemaitre-Robertson-Walker model within loop quantum cosmology, from the perspective of the two available quantization prescriptions. We focus our attention on the existence of the so-called 'inverse corrections' in the quantization process. We derive the corresponding quantum constraint operators, and study in detail the issue of the self-adjointness of the quantum Hamiltonian constraint for two different quantizations based on open holonomies. Furthermore, we analyse in detail the resulting effective theories, paying special attention to issues such as the boundedness of different scalar observables such as energy density and expansion. We show that the inclusion of the inverse corrections solves one of the main undesirable features of the previous formalism, namely the unboundedness of the energy density and expansion near the bounce on the full (effective) phase space. We briefly comment on possible consequences of this new feature.