  A new quantum divergence induced from the $\alpha-z$ Renyi relative entropy,
called the $\alpha-z$ Bures-Wasserstein quantum divergence, has been recently
introduced. We investigate in this paper properties of the right mean, which is
a unique minimizer of the weighted sum of $\alpha-z$ Bures-Wasserstein quantum
divergences to each points. Many interesting operator inequalities of the right
mean with the matrix power mean including the Cartan mean are presented.
Moreover, we verify the trace inequality with the Wasserstein mean and provide
bounds for the Hadamard product of two right means.
