One-loop electron self-energy for the bound-electron $g$ factor

We report calculations of the one-loop self-energy correction to the bound-electron $g$ factor of the $1s$ and $2s$ states of light hydrogen-like ions with the nuclear charge number $Z \le 20$. The calculation is carried out to all orders in the binding nuclear strength. We find good agreement with previous calculations and improve their accuracy by about two orders of magnitude.