Integral Inequality for the Polar Derivative of a Polynomial with the Roots Separated from the Zero

The classical Erdös–Lax inequality establishes a sharp pointwise estimate for the derivative of a polynomial that does not vanish in the unit disk. Quantitative refinements of this result, depending on the radius of the zero-free disk, were found by Malik, Qazi, and many other authors. Similar results for integral norms were obtained in the works of Dewan et al., and Chanam. In this paper, a refined integral version of the above inequalities is obtained for the case of a polar derivative of a polynomial. Moreover, the obtained result, in turn, extends and generalizes the theorems of Gardner et al., Dewan et al., and Chanam.