Small values of polynomials and potentials with $L_p$ normalization

Abstract:

For a polynomial $P$ of degree $\leq n$, normalized by the condition \[ \frac 1{2\pi }\int _0^{2\pi }\mid P(re^{i\theta })\mid ^pd\theta =1, \] we show that $E(P;r;\varepsilon ):=\{z:\mid z\mid \leq r,\mid P(z)\mid \leq \varepsilon ^n\}$ has $cap$ at most $r\varepsilon \kappa _{np}$, where $\kappa _{np}\leq 2$ is explicitly given and sharp for each $n,r$. Similar estimates are given for other normalizations, such as $p=0$, 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.