Volume 27, Issue 2
On Extremal Properties for the Polar Derivative of Polynomials

Anal. Theory Appl., 27 (2011), pp. 150-157.

Published online: 2011-04

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• Abstract

If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved[5] that $$\max_{|z|=1}|p′(z)| \leq\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu \leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

• Keywords

polynomial, zeros, inequality, polar derivative.

30A10, 30C10, 30C15

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@Article{ATA-27-150, author = {}, title = {On Extremal Properties for the Polar Derivative of Polynomials}, journal = {Analysis in Theory and Applications}, year = {2011}, volume = {27}, number = {2}, pages = {150--157}, abstract = {

If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved[5] that $$\max_{|z|=1}|p′(z)| \leq\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu \leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

}, issn = {1573-8175}, doi = {https://doi.org/10.1007/s10496-011-0150-3}, url = {http://global-sci.org/intro/article_detail/ata/4588.html} }
TY - JOUR T1 - On Extremal Properties for the Polar Derivative of Polynomials JO - Analysis in Theory and Applications VL - 2 SP - 150 EP - 157 PY - 2011 DA - 2011/04 SN - 27 DO - http://doi.org/10.1007/s10496-011-0150-3 UR - https://global-sci.org/intro/article_detail/ata/4588.html KW - polynomial, zeros, inequality, polar derivative. AB -

If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved[5] that $$\max_{|z|=1}|p′(z)| \leq\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu \leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

K. K. Dewan & Arty Ahuja. (1970). On Extremal Properties for the Polar Derivative of Polynomials. Analysis in Theory and Applications. 27 (2). 150-157. doi:10.1007/s10496-011-0150-3
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