On the Polar Derivative of a Bicomplex Polynomial

Abstract

In this paper, we introduce the concept of the polar derivative for bicomplex polynomials and develop a bicomplex analogue of Laguerre’s theorem. We derive several key inequalities involving the polar derivative and extend our analysis to the t-th order polar derivative. Additionally, we explore the relationship between the norms of a bi complex polynomial’s polar derivative and those of its inverse polynomial. In particular, we establish Bernstein-type inequalities for the polar derivative of bicomplex polynomials, offering new insights into their analytic behavior.