  In 1939 P. Tur\'an started to derive lower estimations on the norm of the
derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and
$D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), 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 $n:=\mathop{\mathrm{deg}}
p$ tending to infinity, the precise growth order of the minimal possible
derivative norm is $\sqrt{n}$ for $I$ and $n$ for $D$.
  J. Er\H{o}d continued the work of Tur\'an considering other domains. Finally,
a decade ago the growth of the minimal possible $\infty$-norm of the derivative
was proved to be of order $n$ for all compact convex domains.
  Although Tur\'an himself gave comments about the above oscillation question
in $L^q$ 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 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.
