Location of the zeros of a polynomial relative to certain disks

Abstract:

The zeros of the complex polynomial $P(z) = {z^n} + \Sigma {\alpha _i}{z^{n - 1}}$ are studied under the assumption that some $|{\alpha _k}|$ is large in comparison with the other $|{\alpha _i}|$. It is shown under certain conditions that $P(z)$ has $n - k$ zeros in $|z| \leq {m_ - }$ and $k$ zeros in $|z| \geq {m_ + }$, where ${m_ - } < {m_ + } \leq |{\alpha _k}{|^{1/k}}$; and under suitably strengthened conditions, one of the $k$ zeros of larger modulus is shown to lie in each of the $k$ disks $|z - {( - {\alpha _k})^{1/k}}| \leq R$, where ${m_ - } + R < |{\alpha _k}{|^{1/k}}$.