IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES FOR UNITARILY INVARIANT NORMS

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar and A,B,X is an element of M-n(C): vertical bar vertical bar vertical bar Lambda(nu) X B1-nu vertical bar Lambda(1-nu) X beta(nu)/2 vertical bar vertical bar vertical bar <= (4ro - 1 vertical bar vertical bar vertical bar A(1/2) X B1/2 + 2(1 - 2r(0)) vertical bar vertical bar vertical bar (1 - alpha)A(1/2) X B-1/2 + alpha(AX + XB/2)vertical bar vertical bar vertical bar, where 1/4 <= nu 3/4, alpha is an element of [1/2, infinity) and r(0) = min{nu, 1 - nu}. (C) 2020 Mathematical Institute Slovak Academy of Sciences.