Some integral inequalities for the polar derivative of a polynomial

Abstract

If P(z) is a polynomial of degree n which does not vanish in |z| < 1, then it is recently proved by Rather [Jour. Ineq. Pure and Appl. Math., 9 (2008), Issue 4, Art. 103] that for every γ> 0 and every real or complex number α with |α| ≥ 1,

$\begin{gathered} \left\{ {\int_0^{2\pi } {\left| {D_\alpha P(e^{i\theta } )} \right|^\gamma d\theta } } \right\}^{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \leqslant n(|\alpha | + 1)C_\gamma \left\{ {\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^\gamma d\theta } } \right\}^{{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} , \hfill \\ C_\gamma \left\{ {\frac{1} {{2\pi }}\int_0^{2\pi } {\left| {1 + e^{i\beta } } \right|^\gamma d\beta } } \right\}^{ - {1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \hfill \\ \end{gathered} $

where D _α P(z) denotes the polar derivative of P(z) with respect to α. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 (1988), 306–313] as a special case.