The Bernstein inequality for slice regular polynomials

TL;DR

This paper extends classical polynomial inequalities to quaternionic slice regular polynomials, providing new proofs and results using harmonic analysis tools, and establishing Turan inequalities in this setting.

Contribution

It offers an alternative proof of the Bernstein inequality for quaternionic polynomials and extends the Ankeny-Rivlin result to quaternions, along with new Turan inequalities.

Findings

Bernstein inequality is valid for quaternionic slice regular polynomials.

Extension of Ankeny-Rivlin result to quaternionic setting.

New Turan inequalities for slice regular polynomials.

Abstract

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.