Numerical solution of the omitted area problem of univalent function theory
The omitted area problem was posed by Goodman in 1949: what is the maximum area $A^*$ of the unit disk D that can be omitted by the image of the unit disk under a univalent function normalized by f(0)=0 and f'(0)=1? The previous best bounds were 0.240005$\pi$ < $A^*$ < .31$\pi$. H...
| Main Authors: | , |
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| Format: | Report |
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Unspecified
2001
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| _version_ | 1797053314018312192 |
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| author | Banjai, L Trefethen, L |
| author_facet | Banjai, L Trefethen, L |
| author_sort | Banjai, L |
| collection | OXFORD |
| description | The omitted area problem was posed by Goodman in 1949: what is the maximum area $A^*$ of the unit disk D that can be omitted by the image of the unit disk under a univalent function normalized by f(0)=0 and f'(0)=1? The previous best bounds were 0.240005$\pi$ < $A^*$ < .31$\pi$. Here the problem is addressed numerically and it is found that these estimates are slightly in error. To ten digits, the correct value appears to be $A^*$ = 0.2385813248$\pi$. |
| first_indexed | 2024-03-06T18:42:06Z |
| format | Report |
| id | oxford-uuid:0d3832e1-1677-4e83-8f5a-f0464e5f8b27 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:42:06Z |
| publishDate | 2001 |
| publisher | Unspecified |
| record_format | dspace |
| spelling | oxford-uuid:0d3832e1-1677-4e83-8f5a-f0464e5f8b272022-03-26T09:39:20ZNumerical solution of the omitted area problem of univalent function theoryReporthttp://purl.org/coar/resource_type/c_93fcuuid:0d3832e1-1677-4e83-8f5a-f0464e5f8b27Mathematical Institute - ePrintsUnspecified2001Banjai, LTrefethen, LThe omitted area problem was posed by Goodman in 1949: what is the maximum area $A^*$ of the unit disk D that can be omitted by the image of the unit disk under a univalent function normalized by f(0)=0 and f'(0)=1? The previous best bounds were 0.240005$\pi$ < $A^*$ < .31$\pi$. Here the problem is addressed numerically and it is found that these estimates are slightly in error. To ten digits, the correct value appears to be $A^*$ = 0.2385813248$\pi$. |
| spellingShingle | Banjai, L Trefethen, L Numerical solution of the omitted area problem of univalent function theory |
| title | Numerical solution of the omitted area problem of univalent function theory |
| title_full | Numerical solution of the omitted area problem of univalent function theory |
| title_fullStr | Numerical solution of the omitted area problem of univalent function theory |
| title_full_unstemmed | Numerical solution of the omitted area problem of univalent function theory |
| title_short | Numerical solution of the omitted area problem of univalent function theory |
| title_sort | numerical solution of the omitted area problem of univalent function theory |
| work_keys_str_mv | AT banjail numericalsolutionoftheomittedareaproblemofunivalentfunctiontheory AT trefethenl numericalsolutionoftheomittedareaproblemofunivalentfunctiontheory |