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 $\phi$ of a power series $f(z)=1+{ \sum _{n=1}^{\infty }} a_{i}z^{i}$ with 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 computing the optimal root when $\phi /2\pi$ is rational. When $g\geq 2\sqrt {2}+3$ we show that the smallest disc containing two roots has radius $(\sqrt {g}+1)^{-1}$ 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 $g$. 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, $\lambda$, of these pseudo-beta-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 $|\lambda |\leq 3/2$.