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P. Tur\'an was the first to derive lower estimations on the uniform norm ofthe derivatives of polynomials $p$ of uniform norm $1$ on the intervalI:=[-1,1] and the disk D:=$\{z \in C~:~|z| \le 1\}$, under the normalizationcondition that the zeroes of the polynomial p in question all lie in I or D,resp. Namely, in 1939 he proved that with n:=deg p tending to infinity, theprecise growth order of the minimal possible derivative norm is $\sqrt{n}$ forI and n for D.  Already the same year J. Er\H{o}d considered the problem on other domains. Inhis most general formulation, he extended Tur\'an's order n result on D to acertain general class of piecewise smooth convex domains. Finally, a decade agothe growth order of the minimal possible norm of the derivative was proved tobe n for all compact convex domains.  Tur\'an himself gave comments about the above oscillation question in $L^q$norm on D. Nevertheless, till recently results were known only for I, D andso-called R-circular domains.  Continuing our recent work, also here we investigate the Tur\'an-Er\H{o}dproblem on general classes of domains.

Tur\'an type converse Markov inequalities in $L^q$ on a generalized Er\H{o}d class of convex domains