Entropy relations and bounds of BTZ black hole in gravity's rainbow

In this paper, we present the entropy relations and bounds of Banados-Teitelboim-Zanelli (BTZ) black hole in two models of gravity's rainbow. Because of the effect of gravity's rainbow, one can find that the entropy product and sum both lost their universality and become mass-dependent. On the other hand, comparing the entropy bound of event horizon to the BTZ case, it is shown that the angular momentum J enlarges the entropy hound while the gravity's rainbow parameter gamma diminishes it. For the entropy bound of Cauchy horizon, the gravity's rainbow parameter gamma enlarges it at the large gamma limit, while gamma diminishes it at the small gamma limit. These suggest some clues on the geometrical origin of black hole entropy bounds.