Bernstein-type Inequalities for Bicomplex Polynomials

Abstract

This paper considers the well-known Bernstein and Erdős–Lax inequalities in the case of bicomplex polynomials. We shall prove that the validity of these inequalities depends on the norm in use and we consider the cases of the Euclidean, Lie, dual Lie and hyperbolic-valued norms. In particular, we show that in the case of the Euclidean norm the inequalities holds keeping the same relation with the degree of the polynomial that holds in the classical complex case, but we have to enlarge the radius of the ball. In the case of the dual Lie norm both the relation with the degree and the radius of the ball have to be changed. Finally, we prove that the exact analogs of the two inequalities hold when considering the Lie norm and the hyperbolic-valued norm. In the case of these two norms we also show the validity of the maximum modulus principle for bicomplex holomorphic functions.

This paper is dedicated to our friend and colleague Daniel Alpay on the occasion of his 60th birthday.

Mathematics Subject Classification (2000). 30G35, 30C10