Self-energy screening effects in the $g$ factor of Li-like ions

We report an investigation of the self-energy screening effects for the $g$ factor of the ground state of Li-like ions. The leading screening contribution of the relative order $1/Z$ is calculated to all orders in the binding nuclear strength parameter $Zalpha$ (where $Z$ is the nuclear charge number and $alpha$ is the fine-structure constant). We also extend the known results for the $Zalpha$ expansion of the QED screening correction by deriving the leading logarithmic contribution of order $alpha^5lnalpha$ and obtaining approximate results for the $alpha^5$ and $alpha^6$ contributions. The comparison of the two approaches yields a stringent check of consistency of the two calculations and allows us to obtain improved estimations of the higher-order screening effects.