Silverman proved a height inequality for jointly regular family of rational
maps and the author improved it for jointly regular pairs. In this paper, we
provide the same improvement for jointly regular family; if S is a jointly
regular set of rational maps, then

\sum_{f\in S} \dfrac{1}{\deg f} h\bigl(f(P) \bigr) > (1+ \dfrac{1}{r}) f(P) -
C

where r = \max_{f\in S} r(f).