On the number of real critical points of logarithmic derivatives and the Hawaii conjecture
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arXiv
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| Format: | Preprint |
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2009
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| _version_ | 1866918073669255168 |
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| author | Tyaglov, Mikhail |
| author_facet | Tyaglov, Mikhail |
| contents | For a given real entire function $ϕ$ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions $Q=(ϕ'/ϕ)'$ and $Q_1=(ϕ''/ϕ')'$. This connection leads to a proof of the Hawaii conjecture [T.Craven, G.Csordas, and W.Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405--431] stating that the number of real zeros of $Q$ does not exceed the number of nonreal zeros of $ϕ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_0902_0413 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | On the number of real critical points of logarithmic derivatives and the Hawaii conjecture Tyaglov, Mikhail Classical Analysis and ODEs Complex Variables 30C15, 26C05, 26C10, 26C15, 12D10, 30C10, 26E05 For a given real entire function $ϕ$ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions $Q=(ϕ'/ϕ)'$ and $Q_1=(ϕ''/ϕ')'$. This connection leads to a proof of the Hawaii conjecture [T.Craven, G.Csordas, and W.Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405--431] stating that the number of real zeros of $Q$ does not exceed the number of nonreal zeros of $ϕ$. |
| title | On the number of real critical points of logarithmic derivatives and the Hawaii conjecture |
| topic | Classical Analysis and ODEs Complex Variables 30C15, 26C05, 26C10, 26C15, 12D10, 30C10, 26E05 |
| url | https://arxiv.org/abs/0902.0413 |