Bernstein inequalities for quaternionic polynomials in the setting of generalized polynomials

Abstract

Recently, several authors proved a Bernstein type inequality for so called quaternionic unilateral polynomials. Although these polynomials are not generalized complex polynomials between normed spaces, it will be proved in this contribution that their restrictions to each complex plane satisfy a Bernstein inequality for the real Fréchet derivative. This result will imply Bernstein's inequality for quaternionic unilateral polynomials which in addition holds for all complex norms in which the unit closed ball contains the real unit quaternion. The second main contribution consists in providing Bernstein-type inequalities for so called quaternionic canonical generalized polynomials. These polynomials have a factorized form which, due to the lack of commutativity for quaternionic multiplication leads to a completely different type of investigation comparing to the other class of quaternionic unilateral polynomials. However, Bernstein's inequality is proved for some remarkable classes of such polynomials of degree less than or equal to 3 and for a class of polynomials of arbitrary degree having at most two distinct roots.