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We commence the evaluation of the one- and two-loop binding corrections tothe $g factor for an electron in a hydrogenlike system of order alpha^2 (Zalpha)^5 and consider diagrams with closed fermion loops. The one-loopvacuum-polarization correction is rederived and confirmed. For the two-loopvacuum-polarization correction, due to a specific gauge-invariant set ofdiagrams 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 coefficientsinferred from the gauge-invariant subset, we obtain a numerically largetentative estimate for the complete two-loop binding correction to the g factor(sum of self-energy and vacuum polarization).

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