Nikol'skii-Stechkin inequality for trigonometric polynomials in \(L_0\) (Q881118)

Let \(P_n(z),z\in \mathbb{C}\), be an algebraic polynomial of degree \(n\) with complex coefficients. The ''zero norm'' or the Mahler measure of \(P_n\) is defined as follows \[ \| P_n\| _0=\exp\left(\frac{1}{2\pi}\int^\pi_{-\pi} \ln| P_n(e^{it})| dt\right). \] The author proves a generalization of the Bernshtein-Nikol'skiĭ--Stechkin inequality for the ''0-norm'': \[ \| P_n'\| _0\leq A_{n,h}\| P_n(z)-P_n(ze^{ih})\| _0,\quad 0<h<\frac{2\pi}{n}, \] and it is shown that this inequality is sharp. Here \(A_{n,h}\) is the ''0-norm'' of a certain function, and it is shown that \[ A_{n,h}\leq\frac{n}{\sin(\frac{nh}{2})}\| \sup_{0\leq t\leq 1}| 1+tz| ^{n-1}\| _0, \] where the second factor is equivalent to \(2^{\frac{n-1}{2}}\) as \(n\to\infty\). A similar result is established for trigonometric polynomials.