Abstract
Consider any complex polynomial pn(z)=1+\(\mathop \Sigma \limits_{j = l}^n\) ajzj which satisfies \(\mathop \Sigma \limits_{j = l}^n\)|aj|=1, and let Γn denote the supremum of the minimum moduli on |z|=1 of all such polynomials pn(z). We show that
If the coefficients of pn(z) are further restricted to be positive numbers and if \(\tilde \Gamma _n\) denotes the analogous supremum of the minimum modulion |z|=1 of such polynomials, we similarly show that
We also include some recent numerical experiments on the behavior of Γn, as well as some related conjectures.
Research supported by the Air Force Office of Scientific Research, the Department of Energy, and the Alexander von Humboldt Stiftung.
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Ruscheweyh, S., Varga, R.S. (1984). On the minimum moduli of normalized polynomials. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072408
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DOI: https://doi.org/10.1007/BFb0072408
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