Partial sums of starlike and convex functions

Let f(n)(z) = z + Sigma(k = 2)(n)a(k)z(k) be the sequence of partial sums of a function f(z) = z + Sigma(k = 2)(infinity)a(k)z(k) that is analytic in \z\ < 1 and either starlike of order alpha or convex of order alpha, 0 less than or equal to alpha < 1. When the coefficients {a(k)} are ''small,'' we determine lower bounds on Re{f(z)/f(n)(z}, Re{f(n)(z)/f(z)}, Re{f'(z)/f(n)'(z)}, and Re{f(n)'(z)/f'(z)}. In all cases, the results are sharp for each n. (C) 1997 Academic Press.