Growth of the Maximum Modulus of Polynomials With Prescribed Zeros
If $p(z)$ be a polynomial of degree $n$ which does not vanish in the disk $\left|z\right|<k$, then for $k=1$, it iswell known that\begin{align*}\max_{\left|z\right|=r<1}\left|p(z)\right|&\geq\left(\frac{r+1}{2}\right)^n\max_{\left|z\right|=1}\left|p(z)\right|,\\\intertext{and} \max_{\left|...
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| Published in | Sarajevo journal of mathematics Vol. 7; no. 1; pp. 11 - 17 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
10.06.2024
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| Online Access | Get full text |
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| Summary: | If $p(z)$ be a polynomial of degree $n$ which does not vanish in the disk $\left|z\right|<k$, then for $k=1$, it iswell known that\begin{align*}\max_{\left|z\right|=r<1}\left|p(z)\right|&\geq\left(\frac{r+1}{2}\right)^n\max_{\left|z\right|=1}\left|p(z)\right|,\\\intertext{and} \max_{\left|z\right|=R>1}\left|p(z)\right|&\leq\frac{R^n+1}{2}\max_{\left|z\right|=1}\left|p(z)\right|.\end{align*}In this paper, we consider a class of lacunary polynomials and present certain generalizations as well as improvements of the above inequalities for the two cases $k\geq 1$ and $k<1$.
2000 Mathematics Subject Classification. 30A10, 30C10, 30C15 |
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| ISSN: | 1840-0655 2233-1964 |
| DOI: | 10.5644/SJM.07.1.02 |