Turán–Erőd type converse Markov inequalities on general convex domains of the plane in the boundary $L^q$ norm

In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm $1$ on $\mathbb I:=[-1,1]$ (interval) and $\mathbb D:=\{z\in \mathbb C: |z|\le 1\}$ (disk) under the normalization condition that the zeroes of the polynomial in question all lie in $\mathbb I$ or $\mathbb D$, respectively. For the maximum norm he found that with $n:=\deg p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt {n}$ for $\mathbb I$ and $n$ for $\mathbb D$. J. Erőd continued the work of Turán considering other domains. Finally, about 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 Turán himself gave comments about the above oscillation question in $L^q$ norms, till recently results were known only for $\mathbb D$ and $\mathbb I$. 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 $n/\kern -1pt\log n$ for all compact convex domains.