Turán—Erőd Type Converse Markov Inequalities on General Convex Domains of the Plane in the Boundary Lq Norm

Resumo

In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on \(\mathbb{I}:= [-1,1]\) (interval) and \(\mathbb{D}:= \{z \in \mathbb{C}: |z| \leq 1\}\) (disk) under the normalization condition that the zeroes of the polynomial in question all lie in \(\mathbb{I} \) or \(\mathbb{D} \), respectively. For the maximum norm he found that with n:= deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √n for \(\mathbb{I} \) and n for \(\mathbb{D} \). J. Erőd continued the work of Turán considering other domains. Finally, about a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in L^q norms, till recently results were known only for \(\mathbb{D} \) and \(\mathbb{I} \). Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in L^q norm the oscillation order is at least n/log n for all compact convex domains.