Analogue of Gauss-Lucas theorem for non convex set on the complex plane

Bl. Sendov

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arXiv · 1402.6425

Analogue of Gauss-Lucas theorem for non convex set on the complex plane

Bl. Sendov

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Abstract

Let $S(ϕ)= \{z:\;|\arg(z)|\geq ϕ\}$ be a sector on the complex plane $\CC$. If $ϕ\geq π/2$, 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.

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Bl. Sendov. 2015-02-02. Analogue of Gauss-Lucas theorem for non convex set on the complex plane. https://arxiv.org/abs/1402.6425

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