An inequality for self-inverse polynomials
\textit{N. G. de Bruijn} [Nederl. Akad. Wet., Proc. 50, 1265-1272 (1947; Zbl 0029.19802)] proved the following Theorem. If the n-th degree polynomial p(z) is \(\neq 0\) for \(| z|<1\), then \[ \int^{2\pi}_{0}| p'(e^{i\vartheta})/n|^{\delta}d\vartheta \leq C_{\delta}\int^{2\pi}_{0}| p(e^{i\vartheta})|^{\delta}d\vartheta \] holds for every \(\delta\geq 1\), where \[ C_{\delta}=2\pi /\int^{2\pi}_{0}| 1+e^{i\eta}|^{\delta}d\eta =2^{- \delta}\sqrt{\pi}\Gamma(frac{1}{2}\delta +1)/\Gamma(frac{1}{2}\delta +frac{1}{2}). \] The present authors state that the preceding inequality also holds for self-inverse polynomials (whose zeros lie symmetric with respect to the unit circle). See also \textit{P. J. O'Hara} and \textit{R. S. Rodriguez} [Proc. Am. Math. Soc. 44, 331-335 (1974; Zbl 0258.30002)].
scientific article; zbMATH DE number 3847704