In 1972, D.A. Brannan conjectured that all of the odd coefficients, a 2n+1 , of the power series (1 + xz) α /(1 − z) were dominated by those of the series (1 + z) α /(1 − z) for the parameter range 0 < α < 1, after having shown that this was not true for the even coefficients. He verified the case when 2n + 1 = 3. The case when 2n + 1 = 5 was verified in the mid-eighties by J.G. Milcetich. In this paper, we verify the case when 2n + 1 = 7 using classical Sturm sequence arguments and some computer algebra.
1980
Abstract.Let D denote the open unit disk and let f (z) = ∞ 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 a n+1 → 1, and a 0 am ≤ am b , 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) = ∞ n=0 g 1 n+c z n where g(x) = ∞ n=0 bnx n ≡ 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.
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