  By constructing a new coupling, the log-Harnack inequality is established for
the functional solution of a delay stochastic differential equation with
multiplicative noise. As applications, the strong Feller property and heat
kernel estimates w.r.t. quasi-invariant probability measures are derived for
the associated transition semigroup of the solution. The dimension-free Harnack
inequality in the sense of \cite{W97} is also investigated.
