Power series with restricted coefficients and a root on a given ray

We consider bounds on the smallest possible root with a specified argument φ of a power series f(z) = 1 + ∑∞ n=1aiz i with coefficients ai in the interval [−g, g]. We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when φ/2π is rational. When g ≥ 2√2 + 3 we show that the smallest disc containing two roots has radius ( √ g + 1)−1 coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller g. We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates, λ, of these pseudobeta-numbers either lie inside the unit circle or their reciprocals must be roots of [−1/2, 1/2) power series; in particular we obtain the sharp inequality |λ| ≤ 3/2.