Leírás
Abstract: In 1939 P. Turán started to derive lower estimations on the maximum norm of the derivatives of polynomials of norm 1 on the interval I and the disk D under the normalization condition that the zeroes of the polynomial in question all lie in I or 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 I and n for D. J. Erőd continued the work of Turán considering other domains. Finally, about a decade ago the growth of the minimal possible maximal 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 Lq norms, till recently results were known only for D and I. Recently, we have found order n lower estimations for several general classes of compact convex domains, and proved that in Lq norm the oscillation order is at least n/log n for all compact convex domains.