Some maximal polynomials must be nonreal
Let \(n>1\) and let \(\beta\) be a complex number of modulus not exceeding 1. \(S(n,\beta)\) is the set of complex polynomials of degree \(n\) whose roots all lie in the closed unit disk and at least one root is \(\beta\). For a polynomial \(P\) define \(|P|_\beta\) to be the distance between \(\beta\) and the closest root to \(\beta\) of the derivative \(P'\). Sendov's conjecture (sometimes called the Sendov-Ilyeff conjecture) states that for \(P\in S(n,\beta)\), \(|P|_\beta\leq 1\). The present author introduced and studied extremal polynomials related to the above conjecture. These are polynomials \(P\in S(n,\beta)\) for which \(|P |_\beta\) is maximal. Without loss of generality one can assume that \(0\leq \beta\leq 1\). In the present paper the author proves that if \(P\in S(6,\beta)\) is extremal \(P(z)\) cannot be real. This assertion is a result of the following two results obtained by somewhat lengthy technical estimates. Theorem 1. For all real polynomials \(P\in S(6,\beta)\), \[ |P|_\beta\leq 1- {11\over 30} (1-\beta)+ {1\over 30} (1-\beta)^2 +O(1-\beta)^3. \] Theorem 3. There are polynomials \(P\in S(6, \beta)\) for which \[ |P|_\beta=1- {11\over 30} (1-\beta)+ {49\over 900} (1-\beta)^2 +O(1-\beta)^3. \]
scientific article; zbMATH DE number 1115514