Some bounds for the polar derivative of a polynomial (Q1751318)

Summary: The polar derivative of a polynomial \(p(z)\) of degree \(n\) with respect to a complex number \(\alpha\) is a polynomial \(n p(z) + \left(\alpha - z\right) p'(z)\), denoted by \(D_\alpha p(z)\). Let \(1 \leq R \leq k\). For a polynomial \(p(z)\) of degree \(n\) having all its zeros in \(|z|\leq k\), we investigate a lower bound of modulus of \(D_\alpha p(z)\) on \(|z|= R\). Furthermore, we present an upper bound of modulus of \(D_\alpha p(z)\) on \(|z| = R\) for a polynomial \(p(z)\) of degree \(n\) having no zero in \(|z|< k\). In particular, our results in case \(R = 1\) generalize some well-known inequalities.