On some analogues of converse Bernstein inequalities in \(L_0\) (Q1379856)

Let \(P_n\) be an arbitrary algebraic polynomial of degree \(n\) with complex coefficients and for \(p>0\) denote \(\| P_n\|_p=((1/2\pi\int_{-\pi}^{\pi}| P_n(e^{it})|^pdt)^{1/p}\) and denote \[ \| P_n\|_0=\lim_{p\to 0}\| P_n\|_p=\exp(\frac{1}{2pi}\int_{-\pi}^{\pi}\text{ln}| P_n(e^{it})| dt). \] The author proves that for any \(0<r<1\), \(0<h<2\pi/n\) and \(n\geq 2\) the following inequalities are true: \(\| P_n(rz)-P_n(z)\|_0\leq A(r,n)\frac{1-r^n}{n}\| P_n'\|_0\), \(\| P_n(e^{ih}z)-P_n(z)\|_0\leq B(h,n)\frac{2\sin(nh/2)}{n}n\| P_n'\|_0\). He gives an explicit expression for the constants \(A(n,r)\) and \(B(h,n)\). Theses constants are exact.