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 $\gamma < 0$ and every real or complex number $\alpha$ with $|\alpha| \geq 1,$$$\Big\{\int_0^{2\pi}|D_\alpha P(e^{i\theta})|^\gamma d\theta\Big\}^{1/\gamma}\leq n(|\alpha|+1) C_\gamma\Big\{\int_0^{2\pi}|P(e^{i\theta})|^\gamma d\theta\Big\}^{1/\gamma},$$$$C_\gamma=\Big\{\frac{1}{2\pi}\int_0^{2\pi}|1+e^{i\beta}|^\gamma d\beta\Big\}^{-1/\gamma},$$where $D_\alpha P(z)$ denotes the polar derivative of $P(z)$ with respect to $\alpha$. 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.