An inequality for the norm of a polynomial factor

Details An inequality for the norm of a polynomial factor An inequality for the norm of a polynomial factor

2000-01-24

arxiv.org/abs/math/0001124v1

Mathematics

Complex Variables

9 pages, 2 figures, AMS-LaTeX, to appear in Proc. Amer. Math. Soc

Scientific paper

Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\|q\|_{E} \le C^n \|p\|_E$, where $\|\cdot\|_E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.

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