Turán type converse Markov inequalities in Lq on a generalized Erőd class of convex domains

Abstract

P. Turán was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the disk D:={z∈C:|z|≤1} and the interval I:=[−1,1], under the normalization condition that the zeros of the polynomial p in question all lie in D or I, resp. Namely, in 1939 he proved that with n:=degp tending to infinity, the precise growth order of the minimal possible derivative norm is n for D and n for I. Already the same year J. Erőd considered the problem on other domains. In his most general formulation, he extended Turán's order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Turán himself gave comments about the above oscillation question in Lq norm on D. Nevertheless, till recently results were known only for D, I and so-called R-circular domains. Continuing our recent work, also here we investigate the Turán-Erőd problem on general classes of domains. © 2017 Elsevier Inc.