New norm inequalities for commutators of Hilbert space operators

New norm inequalities for commutators of Hilbert space operators are given. Among other inequalities, it is shown that if $A,$ $ B$ $\in \mathbb{B(H)}$ and there exists a real number $z_{0}$, such that $ \Vert A-z_{0}I\Vert=D_{A} $, then
\begin{eqnarray*} \Vert AB \pm BA^*\Vert \leq 2 D_{A} \Vert B \Vert, \end{eqnarray*}
where ${{D}_{A}}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf }} \left\| A-\lambda I \right\| $. In particular, under some conditions, we prove that
\begin{eqnarray*} \Vert AB\Vert \leq D_{A} \Vert B \Vert, \end{eqnarray*}
which is an improvement of submultiplicative norm inequality. Also, we prove several numerical radius inequalities for products of two Hilbert space operators.