On the lemniscate components containing no critical points of a polynomial except for its zeros (Q476652)

The paper answers a question posed by \textit{T. Sheil-Small} [Complex polynomials. Cambridge: Cambridge University Press (2002; Zbl 1012.30001), n. 10.3.2] in a little bit weaker form. The proved result reads as follows. Theorem. Let \(P\) be a polynomial of degree not exceeding \(n\), and let \(E\) be a connected component of the lemniscate set \(|P(z)| \leq 1\) containing no critical points of the polynomial \(P\) different from its zeros. Then, for any \(z\in E\setminus \{a\}\), \[ \left|\frac{(z - a) P^{\prime}(z)}{P(z)}\right| \leq n, \eqno{(1)} \] where \(a\) is the zero of the polynomial \(P\) belonging to the component \(E\). In (1), the equality is attained for any point \(z\) in the case of the polynomial \(P(z) = c z^n\), \(c\not= 0\).