Certain Estimates of Turán's-type For the Maximum Modulus of the Polar Derivative of a Polynomial

Gradimir V. Milovanović, Abdullah Mir, Abrar Ahmad

We establish some lower bound estimates for the maximum modulus of the polar derivative of a polynomial on the unit disk under the assumption that the polynomial has all zeros in another disk. The obtained results sharpen as well as generalize some estimates of Turán's-type that relate the uniform-norm of the polar derivative and the polynomial.