The goal of this paper is to study the properties of zeros of some special quaternionic polynomials with restricted coefficients, namely coefficients whose real and imaginary components satisfy suitable inequalities. We extend the well-known Enestr?m-Kakeya theorem and its various generalizations from complex to the quaternionic setting. The main tools used to derive the bounds for the zeros of these polynomials are the maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable.

Under consideration are the well-known extremal problems of Bernstein-type which relate the uniform norm between polynomials on the unit disk in the plane. We establish a few new inequalities in both directions for the generalized \( {\mathcal{B}}_{n} \)-operator while accounting for the placement of the zeros of the underlying polynomials. Also, we obtain various estimates for the maximum modulus of a polynomial as well as some inequalities of Erdös–Lax type.