Two-loop QED corrections with closed fermion loops for the bound-electron g factor

Two-loop QED corrections with closed fermion loops are calculated for the 1s bound-electron g factor. Calculations are performed to all orders in the nuclear binding strength parameter Z alpha (where Z is the nuclear charge and alpha is the fine-structure constant) except for the closed fermion loop, which is treated within the free-loop (Uehling) approximation in some cases. Comparison with previous Z alpha-expansion calculations is made and the higher-order remainder of order alpha(2)(Z alpha)(5) and higher is separated out from the numerical results.