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).
Categories: math.NT