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.