Volume 27, Issue 4
Some Integral Inequalities for the Polar Derivative of a Polynomial

Abdullah Mir & Sajad Amin Baba

Anal. Theory Appl., 27 (2011), pp. 340-350.

Published online: 2011-11

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  • 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.

  • Keywords

polar derivative, polynomial, Zygmund inequality, zeros.

  • AMS Subject Headings

30A10, 30C10, 30D15, 41A17

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-27-340, author = {}, title = {Some Integral Inequalities for the Polar Derivative of a Polynomial}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {4}, pages = {340--350}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0340-z}, url = {http://global-sci.org/intro/article_detail/ata/4606.html} }
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 -

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.

Abdullah Mir & Sajad Amin Baba. (1970). Some Integral Inequalities for the Polar Derivative of a Polynomial. Analysis in Theory and Applications. 27 (4). 340-350. doi:10.1007/s10496-011-0340-z
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