2014/02/26 by Bl. Sendov, Sendov, Bl.
Mathematics · Physics and Astronomy · #30C15 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Theories and Applications #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.CV #msc:30C15
paper · pdf · doi:10.48550/arxiv.1402.6425
openalex publication_date 2014/02/26 · arxiv created 2015/02/02 · arxiv updated 2015/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let S(ϕ)= \z: |arg(z)|≥ ϕ\ be a sector on the complex plane \CC. If ϕ≥ π/2, then S(ϕ) is a convex set and, according to the Gauss-Lucas theorem, if a polynomial p(z) has all its zeros on S(ϕ), then the same is true for the zeros of all its derivatives. In this paper is proved that if the polynomial p(z) is with real and non negative coefficients, then the same is true also for ϕ< π/2, when the sector is not a convex set on the complex plane.