POWER-SERIES WITH RESTRICTED COEFFICIENTS AND A ROOT ON A GIVEN RAY

We consider bounds on the smallest possible root with a specified argu ment phi of a power series f(z) = 1 + Sigma(n=1)(infinity)a(i)z(i) wit h coefficients a(i) in the interval [-g, g]. We describe the form that the extremal power series must take and hence give an algorithm for c omputing the optimal root when phi/2 pi is rational. When g greater th an or equal to 2 root 2 + 3 we show that the smallest disc containing I two roots has radius (root g + 1)(-1) coinciding with the smallest d ouble real root possible for such a series. It is clear from our compu tations 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 bri efly discussing variants of the beta-numbers. (where the defining inte ger sequence is generated by taking the nearest integer rather than th e integer part). We show that the conjugates, lambda, of these pseudo- beta-numbers either lie inside the unit circle or their reciprocals mu st be roots of [-1/2, 1/2) power series; in particular we obtain the s harp inequality \lambda\ less than or equal to 3/2.