Three-loop radiative-recoil corrections to hyperfine splitting generated by one-loop fermion factors

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with one-loop radiative photon insertions both in the electron and muon lines. An analytic result for these nonlogarithmic corrections of order $\alpha (Z^2\alpha )(Z\alpha )(m/M){\tilde{E}}_F$ is obtained. This result constitutes a next step in the implementation of the program of reduction of the theoretical uncertainty of hyperfine splitting below 10 Hz.