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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 26, 2015 10:00, Пленарные доклады, Moscow, Steklov Mathematical Institute of RAS
 


On an analogue of Gauss–Lucas theorem for a non-convex set on the complex plane

B. H. Sendov

Bulgarian Academy of Sciences
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B. H. Sendov
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Abstract: Let $S(\phi)= ż:|{\arg(z)}|\geq \phi\}$ be a sector on the complex plane $\mathbb C$. If $\phi\geq \pi/2$, then $S(\phi)$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\phi)$, 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 $\phi < \pi/2$, when the sector is not a convex set.

Materials: abstract.pdf (45.6 Kb)

Language: English

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