  Some 76 years ago P. Tur\'an 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 : |z| <= 1} under the normalization condition that the zeroes of the
polynomial 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 square-root{n} for I and
n for D.
  J. Er\"od continued the work of Tur\'an and extended his results to several
other domains. The growth of the minimal possible ratio of the infinity norm of
the derivative and the polynomial itself was proved to be of order n for all
compact convex domains a decade ago.
  Although Tur\'an himself gave comments about the above oscillation question
in Lq norms, till recently results were known only for D and I. Here we prove
that in Lq norm the oscillation order is again n for a certain class of convex
domains, including all smooth convex domains and also convex polygonal domains
having no acute angles at their vertices.

Classical Analysis and ODEs