On the maximum modulus of a complex polynomial

Year 2022, Volume: 71 Issue: 3, 666 - 681, 30.09.2022

Shabir Malik Ashish Kumar

https://doi.org/10.31801/cfsuasmas.989344

Abstract

In this paper we impose distinct restrictions on the moduli of the zeros of $p\left(z\right)=\sum _{v=0}^n{a}_v{z}^v$ and investigate the dependence of $\parallel p\left(Rz\right)-p\left(\sigma z\right)\parallel$, $R>\sigma \ge 1$ on $M_{\alpha }$ and $M_{\alpha +\pi }$, where $M_{\alpha }=\underset{1\le k\le n}{max}|p\left(e^{i(\alpha +2k\pi )/n}\right)|$ and on certain coefficients of $p\left(z\right)$. This paper comprises several results, which in particular yields some classical polynomial inequalities as special cases. Moreover, the problem of estimating $p(1-\frac{w}{n})$, $0<w\leq$ given $p(1)=0$ is considered.

Keywords

Polynomials, definite integral, zeros, inequalities

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