Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains
Polina Yu. Glazyrina, Szilárd Gy. Révész · Mathematical Inequalities & Applications · 2017
Some 77 years ago P. Turn was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I := [-1,1] and on the unit disk D := {z C : |z| 1} under the normalization condition that the zeroes of the polynomial p all lie in the interval or in the disk, respectively. He proved that with n := deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is n for I and n for D . J. Erd continued the work of Turn and extended his results to several other domains. The growth of the minimal possible ratio of the -norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago.