We prove that for f : ¯CI → ¯CI a rational mapping of the Riemann sphere of degree at least 2 and Ω a simply connected immediate basin of attraction to an attracting fixed point, if |(f n)0 (p)| ≥ Cn3+ξ for constants ξ > 0,C > 0 all positive integers n and all repelling periodic points p of period n in Julia set for f, then a Riemann mapping R : ID → Ω extends continuously to ¯ID and FrΩ is locally connected. This improves a result proved by J. Rivera-Letelier for Ω the basin of infinity for polynomials, and 5 + ξ rather than 3 + ξ.