Some generalizations involving the polar derivative for an inequality of Paul Turán (Q705755)

Let \(p(z)\) be a polynomial of degree \(n\) and for a complex number \(\alpha\), let \[ D_\alpha p(z)=np(z)+(a-z)p'(z) \] denote the polar derivative of the polynomial \(p(z)\) with respect to \(\alpha\). In this paper, the authors obtain inequalities for the polar derivative of a polynomial having all its zeros in \(| z| \leq K\). Their results generalize and sharpen a famous inequality of Turán and some other known results in this direction.