turan type inequalities for a class of polynomials with constraints

Let p(z) = σn v=0 avzv be a polynomial of degree n. let p′(z) and dαp(z) = np(z)+(α−z)p′(z) be the ordinary and polar derivative of p(z) respectively. in this paper some sharp lower bound estimates for the maximal modulus of p′(z) and dαp(z) are established in terms of their degrees, coefficients and maximal modulus of p(z) over unit disk under the assumption that all the zeros of p(z) lie in |z| ≤ k, k ≥ 1.