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 Inequa...

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and sh...

We solve the so-called invariance equation in the class of Heinz means, that is, we give necessary and sufficient conditions for the constants 0 p,q,r 1 such that the identity Hp(Hq(x,y),Hr(x,y)) = Hp(x,y) (x,y ∈ R+) holds true where the Heinz mean Hp is defined for 0 p 1 as Hp(x,y) = xpy1−p + x1−pyp 2 . The Taylor expansion of the Heinz mean is used. Mathematics subject classification (2010): ...

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of C2-functions to leaves of transversally oriented codimension one C2-foliations of Riemannian manifolds. That extends partially Salavessa’s work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri [3] and Barbosa-GomesSilveira [2] about foliations of 3-dimensional Riemannian ...

The purpose of this paper is to introduce the weighted logarithmic mean two accretive matrices. Among obtained results, we present some inequalities about when involved matrices are sectorial Our approach allowed us derive a new matrix which connected Heinz mean.