  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.
