Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910543290302464 |
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| author | Glazyrina, Polina Révész, Szilárd Gy. |
| author_facet | Glazyrina, Polina Révész, Szilárd Gy. |
| contents | In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Turán obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality.
For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Turán type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains Glazyrina, Polina Révész, Szilárd Gy. Complex Variables Classical Analysis and ODEs 41A17 In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family $\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given convex, compact subset $K\Subset {\mathbb C}$. While Turán obtained the first results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~ |z|\le 1\}$, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and $n$ for $D$. It took until 2006 to clarify that all compact convex \emph{domains} (with nonempty interior), follow the pattern of the disk, and admit an order $n$ inequality. For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$ results for various classes of domains. Further, in the generality of all convex, compact domains we could show a $c n/\log n$ lower bound together with an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order $n$ Turán type inequality. Here we prove this for all \emph{polygonal} convex domains and any $0< q <\infty$. |
| title | Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains |
| topic | Complex Variables Classical Analysis and ODEs 41A17 |
| url | https://arxiv.org/abs/2407.18404 |