Self-inversive polynomials of odd degree
Let \(P(z) =\sum_{k=0}^m A_k z^k\). The polynomial \(P(z)\) is called a self-inversial polynomial of degree \(m\) if \(A_m\neq 0\), \(A_{m-k} = \varepsilon \overline{A_k}\) \((k=0, 1, \ldots m) \) with fixed complex \(\varepsilon , | \varepsilon | =1.\) The authors prove the following. Let \(P(z)\) be a self-inversial polynomial of odd degree \(m \geq 3.\) Then the inequality \[ | A_m| \geq \cos \left(\frac{\pi}{2(m+1)} \right) \inf _{c, d \in \mathbb{C}, | d| =1}\sum_{k=0}^m | c A_k -d^{m-k}A_m| \] implies that all zeros of \(P\) are on the unit circle and ones are simple.
scientific article; zbMATH DE number 5277384