  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 $A , B, X\in M_n(\mathbb{C})$ such that $A$ and $B$ are
positive definite and $f$ 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 we obtain a series of refinements of the
Heinz operator inequalities, which were proved by Kittaneh and Krni\'c.
