  We present the first analytic computation of the Detweiler-Barack-Sago
gauge-invariant redshift function for a small mass in {\it eccentric} orbit
around a {\it spinning} black hole. Our results give the redshift contributions
that mix eccentricity and spin effects, through second order in eccentricity,
second order in spin parameter, and the eight-and-a-half post-Newtonian order.
