TY - JOUR
T1 - Some Integral Inequalities for the Polar Derivative of a Polynomial
JO - Analysis in Theory and Applications
VL - 4
SP - 340
EP - 350
PY - 2011
DA - 2011/11
SN - 27
DO - http://doi.org/10.1007/s10496-011-0340-z
UR - https://global-sci.org/intro/article_detail/ata/4606.html
KW - polar derivative, polynomial, Zygmund inequality, zeros.
AB - <p style="text-align: justify;">If $P(z)$ is a polynomial of degree $n$ which does not vanish in $|z|&lt;1$, then it is recently proved by Rather [Jour. Ineq. Pure and Appl. Math., 9 (2008), Issue 4, Art. 103] that for every $\gamma &lt; 0$ 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.</p>