  We consider three-loop radiative-recoil corrections to hyperfine splitting in
muonium. These corrections are enhanced by the large logarithm of the
electron-muon mass ratio. The leading logarithm cubed and logarithm squared
contributions were obtained a long time ago. We calculate the
single-logarithmic and nonlogarithmic contributions of order $\alpha^3(m/M)E_F$
generated by gauge invariant sets of diagrams with one- and two-loop
polarization insertions in diagrams with two exchanged photons and radiative
photons, and by diagrams with one-loop radiative photon insertions both in the
electron and muon lines. The results of this paper constitute a next step in
the implementation of the program of reduction of the theoretical uncertainty
of hyperfine splitting below 10 Hz. They improve the theory of hyperfine
splitting, and affect the value of the electron-muon mass ratio extracted from
experimental data on the muonium hyperfine splitting.
