Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Small values of polynomials and potentials with normalization

Author(s): D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 127 (1999), 529-536.
MSC (1991): Primary 30C10, 31A15; Secondary 41A17, 41A44, 30C85
MathSciNet review: 1468199
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Abstract: For a polynomial of degree $\leq n$ , normalized by the condition

$\begin{displaymath}\frac 1{2\pi }\int _0^{2\pi }\mid P(re^{i\theta })\mid ^pd\theta =1, \end{displaymath}$

we show that $E(P;r;\varepsilon ):=\{z:\mid z\mid \leq r,\mid P(z)\mid \leq \varepsilon ^n\}$ has at most $r\varepsilon \kappa _{np}$ , where $\kappa _{np}\leq 2$ is explicitly given and sharp for each . Similar estimates are given for other normalizations, such as , and for planar measure, and for generalized polynomials and potentials, thereby extending work of Cuyt, Driver and the author for $p=\infty$ . The relation to Remez inequalities is briefly discussed.

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