Extremal Problems for a Polynomial and Its Polar Derivative

This paper considers the well known Erdős–Lax and Turán-type inequalities that relate the uniform norm of a univariate complex coefficient polynomial to that of its derivative on the unit circle in the plane. Here, we establish some new inequalities that relate the uniform norm of a polynomial and its polar derivative while taking into account the placement of the zeros and the extremal coefficients of the polynomial. The obtained results strengthen some recently proved Erdős–Lax and Turán-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in the literature on this subject.