Upper bound estimates for the polar derivative of a polynomial with restricted zeros

Abstract

The polar derivative of a polynomial P(z) of degree n with respect to a complex number ? is a polynomial nP(z) + (??z)P?(z) of degree at most n?1 and is denoted by D?P(z). We consider the class of polynomials P(z) = a0 + ?n v=? avzv, ? ? 1, of degree n such that P(z) ?0 in |z| < k, k ? 1 and establish some upper bound estimates for the maximum modulus of D?P(z) on the unit disk by involving some of the coefficients of P(z). The obtained results refine and generalize some well known polynomial inequalities.