Zeros of Cesàro Sum Approximations

Let D denote the open unit disk and let f(z) = P∞ n=0 anz n be analytic on D with positive monotone decreasing coefficients an. We answer several questions posed by J. Cima on the location of the zeros of polynomial approximants which he originally posed about outer functions. In particular, we show that the zeros of Cesàro approximants to f are well-behaved in the following sense: (1) if an an+1 → 1, and a0 am ≤ amb, then ∂D is the only accumulation set for the zeros of the Cesàro sums of f ; and (2) if f has a representation f(z) = P∞ n=0 g 1 n+c zn where g(x) = P∞ n=0 bnx n 6≡ 0, bn ≥ 0, then we give sufficient conditions so that the convex hull of the zeros of the Cesàro sums of f will contain D .