The matrix Heinz mean and related divergence

In this paper, we introduce a new quantum divergence$$\Phi (X,Y) = \Tr \left[\left(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right)X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$where $0< \alpha <1$.We study the least square problem with respect to divergence. We also show that divergence satisfies Data Processing Inequality in information theory. addition, matrix $p$-power mean $\mu_p(t, A, B) ((1-t)A^p + tB^p)^{1/p}$ in-betweenness property