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In 1939 P. Tur\'an started to derive lower estimations on the norm of thederivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and$D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization conditionthat 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 possiblederivative 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 derivativewas proved to be of order $n$ for all compact convex domains.  Although Tur\'an himself gave comments about the above oscillation questionin $L^q$ norms, till recently results were known only for $D$ and $I$.Recently, we have found order $n$ lower estimations for several general classesof compact convex domains, and conjectured that even for arbitrary convexdomains 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 convexdomains.

Tur\'an-Er\H{o}d type converse Markov inequalities on general convex domains of the plane in $\boldsymbol{L^q}$