Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk
Let the function $\varphi$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $\varphi$, $$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|...
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| creator | Arango, Juan Arbeláez, Hugo Mejía, Diego |
| description | Let the function $\varphi$ be holomorphic in the unit disk $\mathbb{D}$ of
the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset
\mathbb{D}$. We study the level sets and the critical points of the hyperbolic
derivative of $\varphi$,
$$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In
particular, we show how the Schwarzian derivative of $\varphi$ reveals the
nature of the critical points. |
| doi_str_mv | 10.48550/arxiv.1912.03190 |
| format | Article |
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the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset
\mathbb{D}$. We study the level sets and the critical points of the hyperbolic
derivative of $\varphi$,
$$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In
particular, we show how the Schwarzian derivative of $\varphi$ reveals the
nature of the critical points.</description><identifier>DOI: 10.48550/arxiv.1912.03190</identifier><language>eng</language><subject>Mathematics - Complex Variables</subject><creationdate>2019-12</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1912.03190$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1912.03190$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Arango, Juan</creatorcontrib><creatorcontrib>Arbeláez, Hugo</creatorcontrib><creatorcontrib>Mejía, Diego</creatorcontrib><title>Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk</title><description>Let the function $\varphi$ be holomorphic in the unit disk $\mathbb{D}$ of
the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset
\mathbb{D}$. We study the level sets and the critical points of the hyperbolic
derivative of $\varphi$,
$$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In
particular, we show how the Schwarzian derivative of $\varphi$ reveals the
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the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset
\mathbb{D}$. We study the level sets and the critical points of the hyperbolic
derivative of $\varphi$,
$$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In
particular, we show how the Schwarzian derivative of $\varphi$ reveals the
nature of the critical points.</abstract><doi>10.48550/arxiv.1912.03190</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Complex Variables |
| title | Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk |
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