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 $L^{p} (1\leq p \leq+\infty)$ for slice regular polynomials by the Fej\'er 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.