  The relativistic perihelion precession due to the three-body interaction is
derived. We consider a hierarchical coplanar three-body system, such as the
Sun, Jupiter and Saturn, in which both the secondary object as the largest
planet corresponding to Jupiter (mass $m_2$) and the third one corresponding to
Saturn (mass $m_3$) orbit around the primary object corresponding to Sun (mass
$m_1 \gg m_2 \gg m_3$), where the mean orbital radius of the third body is
larger than that of the secondary one (denoted as $\ell$). We investigate the
post-Newtonian effects on the motion of the third body (semimajor axis a,
eccentricity e for the Keplerian orbital elements). Under some assumptions with
a certain averaging, the relativistic perihelion precession of the third mass
by the post-Newtonian three-body interaction is expressed as $6 G m_2 \ell^2
c^{-2} a^{-3} n (1+9e^2/16) (1-e^2)^{-3} $, where G and c denote the
gravitational constant and the speed of light, respectively, and the mean
motion for the third body is denoted as $n = 2\pi a^{3/2} G^{-1/2}
(m_1+m_2)^{-1/2}$. For the Sun-Jupiter-Saturn system, it is $7.8 \times
10^{-6}$ arcsec/cy. This is larger than the Lense-Thirring effect by Sun but it
cannot yet explain the recently reported value for the anomalous perihelion
precession of Saturn as $-0.006 \pm 0.002$ arcsec/cy by Iorio (2009) based on
the analyses by Pitjeva with the EPM2008 ephemerides.
