Some \(L^p\) inequalities for the polar derivative of a polynomial (Q5930146)

Let \(p_n(z)\) denote a complex polynomial of degree \(n\). For \(\alpha \in \mathbb C\), let \(D_{\alpha}\{p_n(z)\}\) denote the polar derivative of \(p_n(z)\); that is, \(D_{\alpha}\{p_n(z)\}=np_n(z)+(\alpha-z)p_n'(z)\). In the paper under review, the authors establish a number of \(L^p\) inequalities for the polar derivative of \(p_n(z)\). These inequalities generalize the results of \textit{A. Aziz} [J. Approximation Theory 55, No. 2, 183-193 (1988; Zbl 0685.41013)] and \textit{N. G. de Bruijn} [Proc. Akad. Wet. Amsterdam 50, 1265-1272 (1947; Zbl 0029.19802)].