Result: On Sendov′s Conjecture for Roots Near the Unit Circle: On Sendov's conjecture for roots near the unit circle

The well known Iliev-Sendov conjecture states that, if \(P\) is a polynomial with complex coefficients all of whose zeros lie in the unit disk then every disk of radius 1 with centre a zero of \(P\) contains a zero of \(P'\). This has now been settled in general only for polynomials of degree \(\leq 6\) [\textit{E. S. Katsoprinakis}, Bull. Lond. Math. Soc. 24, 449-455 (1992)], and for those with at most 5 distinct zeros [\textit{S. Kumar} and \textit{B. G. Shenoy}, J. Math. Anal. Appl. 171, 595-600 (1992; Zbl 0773.30004)]. Here the author considers the class \(S(n,\beta)\) of polynomials of degree \(n\) with all roots in the unit disk and at least one zero at \(\beta\). He denotes by \(| P|_ \beta\) the distance between \(\beta\) and the nearest root of \(P'\). He proves that there are constants \(K_ n\) with \(K_ n\to 1/3\) such that if \(\beta\) is sufficiently close to 1 and \(P\in S(n+1,\beta)\), then \(| P|_ \beta\leq 1-K_ n(1-\beta)\). Thus the Iliev-Sendov conjecture is true in a rather strong sense for roots sufficiently close to the unit circle.