The Bernstein Inequality for Slice Regular Polynomials

Due to the invalidation of the Gauss–Lucas type result for quaternionic polynomials, we first give in this paper an alternative proof of the Bernstein inequality in Lp(1≤p≤+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p} (1\le p \le +\infty )$$\end{document} for slice regular polynomials by the Fejér kernel and the Minkowski inequality. Secondly, we extend a result of Ankeny–Rivlin to the quaternionic setting via the Hopf lemma. By the way, some Turan inequalities are established for slice regular polynomials.