Location of the zeros of polynomials with a prescribed norm

Rahman, Q. I.; Schmeisser, G.

doi:10.1090/S0002-9947-1974-0349968-1

Location of the zeros of polynomials with a prescribed norm
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by Q. I. Rahman and G. Schmeisser

Trans. Amer. Math. Soc. 196 (1974), 69-78

DOI: https://doi.org/10.1090/S0002-9947-1974-0349968-1

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Abstract:

For monic polynomials ${f_n}(z)$ of degree n with prescribed ${L^p}$ norm $(1 \leq p \leq \infty )$ on the unit circle or supremum norm on the unit interval we determine bounded regions in the complex plane containing at least $k(1 \leq k \leq n)$ zeros. We deduce our results from some new inequalities which are similar to an inequality of Vicente Gonçalves and relate the zeros of a polynomial to its norm.

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Trigonometric series

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