  We consider the best constant in the Rellich-Hardy inequality (with a radial
power weight) for curl-free vector fields on $\mathbb{R}^N$, originally found
by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This
inequality is considered as an intermediate between Hardy-Leray and
Rellich-Leray inequalities. Under the curl-free condition, we compute the new
explicit best constant in the inequality and prove the non-attainability of the
constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.
