Variation of the canonical height in a family of rational maps

Let d ≥ 2 be an integer, let c ∈ ℚ(t) be a rational map, and let be a family of rational maps indexed by t. For each t = λ ∈ ℚ, we let ĥ_fλ (c(λ)) be the canonical height of c(λ) with respect to the rational map f _λ; also we let ĥ_f (c) be the canonical height of c on the generic fiber of the above family of rational maps. We prove that there exists a constant C depending only on c such that for each λ ∈ ℚ |ĥ_fλ (c(λ)) - ĥ_f(c) · h(λ)| ≤ C. In particular, we show that λ→ ĥ_fλ (c(λ)) is a Weil height on ℙ¹. This improves a result of Call and Silverman, 1993, for this family of rational maps.