The Growth Order of the Optimal Constants in Turán-Erőd Type Inequalities in $$L^q(K,\mu )$$

Abstract In 1939 Turán raised the question about lower estimations of the maximum norm of the derivatives of a polynomial p of maximum norm 1 on the compact set K of the complex plain under the normalization condition that the zeroes of p in question all lie in K . Turán studied the problem for the interval I and the unit disk D and found that with $$n:= \deg p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>deg</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> tending to infinity, the precise growth order of the minimal possible derivative norm (oscillation order) is $$\sqrt{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:math> for I and n for D . Erőd continued the work of Turán considering other domains. Finally, in 2006, Halász and Révész proved that the growth of the minimal possible maximal norm of the derivative is of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in $$L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> 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 proved that in $$L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> norm the oscillation order is at least $$n/\log n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> for all compact convex domains. In the present paper we prove that the oscillation order is not greater than n for all compact (not necessarily convex) domains K and $$L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> norm with respect to any measure supported on more than two points on K .