A proof of Ilieff's conjecture for polynomials with four zeros (Q913013)

The authors give an additional proof of Iliev's conjecture (referred to also as Sendov's conjecture) for polynomials of degree four. This conjecture states that any closed disk of radius unity about a zero of a polynomial of degree n whose zeros all lie in the closed unit disk, contains at least one zero of its derivative. The case \(n=4\) was first established by the reviewer [Pac. J. Math. 26, 159-161 (1968; Zbl 0172.359)]. The case \(n=5\) was established by \textit{A. Meir} and \textit{A. Sharma} [Pac. J. Math. 31, 459-469 (1969; Zbl 0194.102)]. The case \(n=6\) is settled by Katsoprinakis (to appear). In addition, the boundary case of a zero on the unit circumference was first proved by the reviewer ibid. and refined by \textit{A. W. Goodman}, \textit{Q. I. Rahman} and \textit{J. S. Ratti} [Proc. Am. Math. Soc. 21, 273- 274 (1969; Zbl 0172.359)]. Other partial results have been established by various authors. For a review of these results consult the articles by \textit{M. Marden} [Am. Math. Monthly 90, 267-276 (1983; Zbl 0535.30010)] and \textit{B. D. Bojanov}, \textit{Q. I. Rahman} and \textit{J. Szynal} [Delay equations, approximation and application, Int. Symp. Mannheim/Ger. 1984, ISNM 74, 83-93 (1985; Zbl 0577.30006)].