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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2505.13285 |
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| _version_ | 1866909615900327936 |
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| author | Komarov, Mikhail A. |
| author_facet | Komarov, Mikhail A. |
| contents | Let $K\subset \mathbb{C}$ be a convex compact set, and let $Π_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Turán type inverse Markov factor is defined by $M_n(K)=\inf_{P\in Π_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound $M_n(K)\ge c\left(wn/d^2+\sqrt{n}/d\right)$, $c:=0.00015$, where $d>0$ is the diameter of $K$ and $w\ge 0$ is the minimal width (the smallest distance between two parallel lines between which $K$ lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28\left(wn/d^2+\sqrt{n}/d\right)$ for all $n,w,d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the precise form of the inverse Markov factor for convex sets Komarov, Mikhail A. Classical Analysis and ODEs 41A17 Let $K\subset \mathbb{C}$ be a convex compact set, and let $Π_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Turán type inverse Markov factor is defined by $M_n(K)=\inf_{P\in Π_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound $M_n(K)\ge c\left(wn/d^2+\sqrt{n}/d\right)$, $c:=0.00015$, where $d>0$ is the diameter of $K$ and $w\ge 0$ is the minimal width (the smallest distance between two parallel lines between which $K$ lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28\left(wn/d^2+\sqrt{n}/d\right)$ for all $n,w,d$. |
| title | On the precise form of the inverse Markov factor for convex sets |
| topic | Classical Analysis and ODEs 41A17 |
| url | https://arxiv.org/abs/2505.13285 |