Extensions of some polynomial inequalities to the polar derivative

Let p ( z ) be a polynomial of degree n and for any real or complex number α, let D α p ( z ) = n p ( z ) + ( α − z ) p ′ ( z ) denote the polar derivative of the polynomial p ( z ) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros inside or outside a circle. Our results shall generalize and sharpen some well-known polynomial inequalities.