On {\(L^p\)} inequalities involving polar derivative of a polynomial
Let \(P(z)\) be a polynomial of degree \(n\). The authors introduce the concept of a polar derivative \(D_\alpha P(z)\) of \(P(z)\) with respect to a complex number \(\alpha\) as follows \[ D_\alpha P(z)= nP(z) + (\alpha-z)P'(z)\,. \] They generalize some results of \textit{N. A. Rather} et al. [Mat. Vesn. 67, No. 1, 56--60 (2015; Zbl 1474.30011)] by proving \(L^p\) inequalities involving the polar derivative of \(P(z)\). Let \(z_m\) be the zeros of \(P(z)\) that satisfy \(| z_m| \geq K_m\geq1\). Then, for any \(p >0\) and for every complex number \(\alpha\) with \(| \alpha| \geq1\), the following inequality holds \[ \left\{\int_0^{2\pi}| D_\alpha\{P(e^{i\theta})\}| ^p\,d\theta\right\}^{1/p}\leq n(| \alpha| +t_0)G_p \left\{\int_0^{2\pi}| P(e^{i\theta})| ^p\,d\theta\right\}^{1/p}\,, \] where the constants \(G_p\) and \(t_0\) depend of \(n\) and \(K_m\). A similar inequality is obtained if \(0<| z_m| \leq K_m\leq1\) and \(| \alpha| \leq1\)
scientific article; zbMATH DE number 6806131