$r$-deformed $\alpha$-$z$-R\'enyi relative entropy

In this article, we consider the $r$-logarithm for defining three-parameter family of Rényi relative entropies that are generalization of the $\alpha$-$z$-Rényi relative entropies.

All the members of $r$-deformed $\alpha$-$z$-Rényi relative entropies satisfy the necessary axioms to be a divergence.

We expose the range of parameters $\alpha$, $z$ and $r$ for which the data processing inequality holds.

We also establish that $r$-deformed $\alpha$-$z$-Rényi relative entropy is an upper bound of the Tsallis relative entropy.

Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $\alpha$-$z$-Rényi relative entropy and the other one, which is discussed in literature.

We investigate the order relationship between these two upper bounds of the Tsallis relative entropy.

We observe that our new upper bound is more tighter when applicable to the density operators.