‎Some improvements of Young inequality and its reverse for positive‎ ‎numbers with Kantorovich constant $K(t‎, ‎2)=frac{(1+t)^2}{4t}$‎ ‎are given‎. ‎Using these inequalities some operator inequalities and‎ ‎Hilbert-Schmidt norm versions for matrices are proved‎. ‎In‎ ‎particular‎, ‎it is shown that if $a‎, ‎b$ are positive numbers and‎ ‎$0 leqslant nu leqslant 1,$ then for all integers $ kgeqsl...

We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. This allows us to prove general results on the monogamy of entanglement and steering quantifiers in continuous variable systems with an arbitrary number of modes ...

Let p1, . . . , pn be positive real numbers. It is well known that for every r < 0 the matrix [(pi + pj) r ] is positive definite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when r > 0. Connections with some other matrices that arise in Loewner’s theory of operator monotone functions and in the theory of spline interpolation are discussed.