REFINED HEINZ OPERATOR INEQUALITIES AND NORM INEQUALITIES

In this article we study the Heinz and Hermite-Hadamard inequalities. We derive the whole series of refinements of these inequalities involving unitarily invariant norms, which improve some recent results, known from the literature. We also prove that if A, B, X is an element of M-n(C) such that A and B are positive definite and f is an operator monotone function on (0, Upsilon). Then vertical bar vertical bar vertical bar f (A)X - X f(B)vertical bar vertical bar vertical bar <= max{vertical bar vertical bar f'(A)vertical bar vertical bar, vertical bar vertical bar f'(B)vertical bar vertical bar}vertical bar vertical bar vertical bar AX - XB vertical bar vertical bar vertical bar. Finally we obtain a series of refinements of the Heinz operator inequalities, which were proved by Kittaneh and Krnic.