  The Detweiler-Barack-Sago redshift function for particles moving along
slightly eccentric equatorial orbits around a Kerr black hole is currently
known up to the second order in eccentricity, second order in spin parameter,
and the 8.5 post-Newtonian order. We improve the analytical computation of such
a gauge-invariant quantity by including terms up to the fourth order in
eccentricity at the same post-Newtonian approximation level. We also check that
our results agrees with the corresponding post-Newtonian expectation of the
same quantity, calculated by using the currently known Hamiltonian for spinning
binaries.
