A Note on the Bernstein Type Inequalities for a Polynomial with Restrictions on Its Zeros

This study examines the well-known inequalities of the type Bernstein and Erdős–Lax that relate a polynomial’s uniform norm to that of its derivative on the unit circle in the plane. Here, we create some new inequalities that, depending on whether some (or all) or none zeros are inside the disc, connect the uniform norm of the polar derivative to the polynomial. The acquired results lead to a number of inequalities that are sharper than the prior ones that are described in the wealth of literature on the subject.