On certain mean values of polynomials on the unit interval.
For any continuous function \(f: [-1,1] \rightarrow {\mathbb C}\) and any \(p \in (0,\infty)\), let \(\| f \|_p\) and \(\| f\|_\infty\) denote the \(L_p\) and the sup norm of \(f\) respectively. It is known [\textit{S. M. Nikolskij}, Tr. Mat. Inst. Steklova 38, 244--278 (1951; Zbl 0049.32301)] that if \(f\) is a polynomial of degree \(n\), then for all \(p>0\), \(\| f \|_\infty \leq C_p n^{2/p} \| f \|_p\) where \(C_p\) is a constant dependent on \(p\) but not on \(n\). Here the order of the magnitude of the bound is the best possible. The authors obtain a sharp version of the foregoing inequality for polynomials not vanishing in the open unit disk. As an application, an extension of an inequality of \textit{P. Erdős} [Ann. Math. (2) 41, 310--313 (1940; Zbl 0024.00403 and JFM 66.0303.01)] is obtained.
scientific article; zbMATH DE number 1389112