In [1], a relative root Nielsen number $N_{rel}(f;;c)$ is introduced which is a homotopy invariant lower bound for the number of roots at $c;{in};Y$ for a map of pairs of spaces $f;:;(X,;A);{ ightarrow};(Y,;B)$. In this paper, we obtain a minimum theorem for $N_{rel}(f;;c)$ under some new assumptions on the spaces and maps which are different from those in [1].