Quantum corrected thermodynamics of nonlinearly charged BTZ black holes in massive gravity’s rainbow

In this paper, we consider three-dimensional massive gravity's rainbow and obtain black hole solutions in three different cases of Born-Infeld, logarithmic, exponential theories nonlinear electrodynamics. We discuss the horizon structure geometrical properties. Then, study thermodynamics these models by considering first-order quantum correction effects, which appears as a logarithmic term entropy. such effects on stability phase transitions. find that, due to corrections, second-order transition happens Born-Infeld models. modified first law presence corrections.