  Given an elliptic curve E over a function field K=Q(T_1,...,T_n), we study
the behavior of the canonical height ^h_(E_w) of the specialized elliptic curve
E_w with respect to the height of w in Q^n. In this paper, we prove that there
exists a uniform non-zero lower bound for the average of the quotient
^h_(E_w)(P_w)/h(w) for all non-torsion P in E(K).
