Let f(z) = z + ∑(n=2)(∞) (a)n(z) (n) be analytic in the unit disk with the second coefficient a2 satisfying |a2| = 2b, 0 ≤ b ≤ 1. Sharp radius of Janowski starlikeness is obtained for functions f whose nth coefficient satisfies |a(n)| ≤ cn + d (c, d ≥ 0) or |a(n)| ≤ c/n (c > 0 and n ≥ 3). Other radius constants are also obtained for these functions, and connections with earlier results are made.