Integral inequality for the polar derivatives of polynomials.

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [ABSTRACT FROM AUTHOR]