Improved Variants of Young, Heinz, and Hölder Inequalities for Matrices

In this talk we deal with a more precise estimates for the matrix versions of Young, Heinz, and Hölder inequalities. First we give an improvement of the matrix Heinz inequality for the case of the Hilbert-Schmidt norm. Then, we refine matrix Young-type inequalities for the case of Hilbert-Schmidt norm, which hold under certain assumptions on positive semidefinite matrices appearing therein. Finally, we give the refinement and the reverse of the matrix Hölder inequality which holds for every unitarily invariant norm. As applications, we also obtain improvements of some well-known matrix inequalities in a quotient form. Our results are compared with the previously known from the literature.