A sharp estimate of area for sublevel-set of Blaschke products
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910545432543232 |
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| author | Kalaj, David |
| author_facet | Kalaj, David |
| contents | Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the sublevel set of $B$ satisfies the following sharp inequality for $t \in [0,1]$: $$|\{z\in \mathbb{D}:|B(z)|<t\}|\le πt^{2/d},$$ with equality at a single point $t\in(0,1)$ if and only if $a_k=0$ for every $k$. In that case the equality is attained for every $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A sharp estimate of area for sublevel-set of Blaschke products Kalaj, David Complex Variables Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the sublevel set of $B$ satisfies the following sharp inequality for $t \in [0,1]$: $$|\{z\in \mathbb{D}:|B(z)|<t\}|\le πt^{2/d},$$ with equality at a single point $t\in(0,1)$ if and only if $a_k=0$ for every $k$. In that case the equality is attained for every $t$. |
| title | A sharp estimate of area for sublevel-set of Blaschke products |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2407.19539 |