Bernstein-type Inequalities for Bicomplex Polynomials

Bernstein-type Inequalities for Bicomplex Polynomials

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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.

ISBN

978-3-319-62362-7

Publication Date

9-2017

Publisher

Springer

City

Cham, Switzerland

Keywords

Bicomplex and hyperbolic numbers, polynomials, Bernstein inequality, Erdős–Lax inequality

Disciplines

Analysis | Numerical Analysis and Computation | Other Mathematics

Comments

In Fabrizio Colombo, Irene Sabadini, Daniele Sgtruppa, and Mihaela Vajiac (Eds.), Advances in Complex Analysis and Operator Theory. Trends in Mathematics.

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Springer

Bernstein-type Inequalities for Bicomplex Polynomials

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