Binding two-loop vacuum-polarization corrections to the bound-electron g factor

We commence the evaluation of the one- and two-loop binding corrections to the g factor for an electron in a hydrogenlike system of order alpha(2)(Z alpha)(5) and consider diagrams with closed fermion loops. The one-loop vacuum-polarization correction is rederived and confirmed. For the two-loop vacuum-polarization correction, due to a specific gauge-invariant set of diagrams with closed fermion loops, we find a correction delta g=7.442(alpha/pi)(2)(Z alpha)(5). Based on the numerical trend of the coefficients inferred from the gauge-invariant subset, we obtain a numerically large tentative estimate for the complete two-loop binding correction to the g factor (sum of self-energy and vacuum polarization).