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

Abstract

We consider bounds on the smallest possible root with a specified argument ϕ \phi of a power series f ( z ) = 1 + ∑ n = 1 ∞ a i z i f(z)=1+{ \sum _{n=1}^{\infty }} a_{i}z^{i} with coefficients a i a_{i} in the interval [ − g , g ] [-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 / 2 π \phi /2\pi is rational. When g ≥ 2 2 + 3 g\geq 2\sqrt {2}+3 we show that the smallest disc containing two roots has radius ( g + 1 ) − 1 (\sqrt {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 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 \lambda , of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of [ − 1 / 2 , 1 / 2 ) [-1/2,1/2) power series; in particular we obtain the sharp inequality | λ | ≤ 3 / 2 |\lambda |\leq 3/2 .