  We prove numerical radius inequalities involving commutators of $G_{1}$
operators and certain analytic functions. Among other inequalities, it is shown
that if $A$ and $X$ are bounded linear operators on a complex Hilbert space,
then \begin{equation*} w(f(A)X+X\bar{f}(A))\leq
{\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast }), \end{equation*} where $A$ is a $G_{1}$
operator with $\sigma (A)\subset \mathbb{D}$ and $f$ is analytic on the unit
disk $\mathbb{D}$ such that $\textrm{{Re}}(f)>0$ and $f(0)=1$.
