Volume 34, Issue 2
Inequalities Concerning the Maximum Modulus of Polynomials

Anal. Theory Appl., 34 (2018), pp. 175-186.

Published online: 2018-07

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• Abstract

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, then for every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh et al. [7] that for $|z|=1$,
$$\min\limits_{|z|=1}\left|P(Rz)+\beta(\frac{R+k}{1+k})^nP(z)\right|\geq k^{-n}\left|R^n+\beta(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$
In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.

• Keywords

Growth of polynomials, minimum modulus of polynomials, inequalities.

30A10, 30C10, 30E15

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@Article{ATA-34-175, author = {}, title = {Inequalities Concerning the Maximum Modulus of Polynomials}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {2}, pages = {175--186}, abstract = {

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, then for every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh et al. [7] that for $|z|=1$,
$$\min\limits_{|z|=1}\left|P(Rz)+\beta(\frac{R+k}{1+k})^nP(z)\right|\geq k^{-n}\left|R^n+\beta(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$
In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2018.v34.n2.7}, url = {http://global-sci.org/intro/article_detail/ata/12585.html} }
TY - JOUR T1 - Inequalities Concerning the Maximum Modulus of Polynomials JO - Analysis in Theory and Applications VL - 2 SP - 175 EP - 186 PY - 2018 DA - 2018/07 SN - 34 DO - http://doi.org/10.4208/ata.2018.v34.n2.7 UR - https://global-sci.org/intro/article_detail/ata/12585.html KW - Growth of polynomials, minimum modulus of polynomials, inequalities. AB -

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, then for every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh et al. [7] that for $|z|=1$,
$$\min\limits_{|z|=1}\left|P(Rz)+\beta(\frac{R+k}{1+k})^nP(z)\right|\geq k^{-n}\left|R^n+\beta(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$
In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.

B. A. Zargar, A. W. Manzoor & Shaista Bashir. (1970). Inequalities Concerning the Maximum Modulus of Polynomials. Analysis in Theory and Applications. 34 (2). 175-186. doi:10.4208/ata.2018.v34.n2.7
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