Refined Heinz operator inequalities and norm inequalities

In this article we study the Heinz and Hermite-Hadamard inequalities. We derive whole series of refinements these inequalities involving unitarily invariant norms, which improve some recent results, known from literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such $A$ $B$ are positive definite $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-Xf(B)|||\leq \max\{||f'(A)||, ||f'(B)||\} |||AX-XB|||. \end{equation*} Finally obtain a inequalities, were proved by Kittaneh Krnic.