  In this paper, we study the symmetrized Talagrand inequality that was proved
by Fathi and has a connection with the Blaschke-Santal\'{o} inequality in
convex geometry. As corollaries of our results, we have several refined
functional inequalities under some conditions. We also give an alternative
proof of Fathi's symmetrized Talagrand inequality on the real line and some
applications.
