Integrated Partial Sums of Convolutions of Univalent Functions

Let S be the set of normalized univalent functions. We show that for f(z) = ∑∞k = 1akzk, g(z) = ∑∞k = 1bkzk ∈ S, [formula] if n < 3 or n > 6. For n = 4 we obtain the corresponding result if g is restricted to the class of close-to-convex functions. The results that we obtain are special cases of a multiplier conjecture for univalent functions as well as an extension of an old result about sections of univalent functions.