On Bernstein's inequality and Kahane's ultraflat polynomials (Q1357297)

Let \(G_n\subset\mathbb{C}[z]\) be the class of unimodular polynomials of degree \(n\) (that is, with coefficients of modulus one). By Parseval's formula for \(P_n\in G_n\), \[ \min_{|z|=1} |P_n(z)|< \sqrt{n+1}< \max_{|z|=1} |P_n(z)|.\tag{1} \] Littlewood conjectured and then Kahane had proved that there is a sequence \(\{P_n\in G_n\}\) such that both the sides of (1) are equal to \(\sqrt{n+1}\) up to \(\varepsilon_n\to 0\) (such \(\{P_n\}\) is called to be \(\{\varepsilon_n\}\)-ultraflat). Three results of the paper give partial confirmations to the following conjecture (Saffaric). If \(\{P_n\}\) is \(\{\varepsilon_n\}\)-ultraflat then \[ \bigl|P_n'\bigr|_q/n\bigl|P_n\bigr|_q= (q+1)^{-1/q}+ O(\varepsilon_n)\quad (0<q<\infty). \] The remaining result states \[ \lim_{n\to\infty} \Biggl(\sup_{P\in G_n} {\bigl|P_n'\bigr|_q\over n\bigl|P_n\bigr|_q}\Biggr)= 1, \] if \(q\neq 2\). The same holds true for \(F_n= G_n\cap \mathbb{R}[z]\). Strikingly, this limit equals \(1/\sqrt 3\), if \(q=2\)!