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Volume 31, Issue 1
Some Integral Mean Estimates for Polynomials with Restricted Zeros

A. Mir, Q. M. Dawood & B. Dar

Anal. Theory Appl., 31 (2015), pp. 81-91.

Published online: 2017-01

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  • Abstract

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$. For $k=1$, it is known that for each $r> 0$ and $|\alpha|\geq 1$, $$n(|\alpha|-1)\Big\{\int_{0}^{2\pi}|P(e^{i\theta})|^{r}d\theta\Big\}^{\frac{1}{r}}\leq \Big\{\int_{0}^{2\pi}|1+e^{i\theta}|^{r}d\theta\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{\alpha}P(z)\big|.$$ In this paper, we shall first consider the case when $k\geq 1$ and present certain generalizations of this inequality. Also for $k\leq 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

  • AMS Subject Headings

30A10, 30C10, 30D15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-31-81, author = {A. Mir, Q. M. Dawood and B. Dar}, title = {Some Integral Mean Estimates for Polynomials with Restricted Zeros}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {1}, pages = {81--91}, abstract = {

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$. For $k=1$, it is known that for each $r> 0$ and $|\alpha|\geq 1$, $$n(|\alpha|-1)\Big\{\int_{0}^{2\pi}|P(e^{i\theta})|^{r}d\theta\Big\}^{\frac{1}{r}}\leq \Big\{\int_{0}^{2\pi}|1+e^{i\theta}|^{r}d\theta\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{\alpha}P(z)\big|.$$ In this paper, we shall first consider the case when $k\geq 1$ and present certain generalizations of this inequality. Also for $k\leq 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n1.7}, url = {http://global-sci.org/intro/article_detail/ata/4624.html} }
TY - JOUR T1 - Some Integral Mean Estimates for Polynomials with Restricted Zeros AU - A. Mir, Q. M. Dawood & B. Dar JO - Analysis in Theory and Applications VL - 1 SP - 81 EP - 91 PY - 2017 DA - 2017/01 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n1.7 UR - https://global-sci.org/intro/article_detail/ata/4624.html KW - Polynomial, zeros, polar derivative. AB -

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$. For $k=1$, it is known that for each $r> 0$ and $|\alpha|\geq 1$, $$n(|\alpha|-1)\Big\{\int_{0}^{2\pi}|P(e^{i\theta})|^{r}d\theta\Big\}^{\frac{1}{r}}\leq \Big\{\int_{0}^{2\pi}|1+e^{i\theta}|^{r}d\theta\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{\alpha}P(z)\big|.$$ In this paper, we shall first consider the case when $k\geq 1$ and present certain generalizations of this inequality. Also for $k\leq 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

A. Mir, Q. M. Dawood and B. Dar. (2017). Some Integral Mean Estimates for Polynomials with Restricted Zeros. Analysis in Theory and Applications. 31 (1). 81-91. doi:10.4208/ata.2015.v31.n1.7
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