Maximal univalent disks of real rational functions and Hermite-Biehler polynomials
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2010
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| _version_ | 1866908425759227904 |
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| author | Kostov, V. Shapiro, B. Tyaglov, M. |
| author_facet | Kostov, V. Shapiro, B. Tyaglov, M. |
| contents | The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1005_1596 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Maximal univalent disks of real rational functions and Hermite-Biehler polynomials Kostov, V. Shapiro, B. Tyaglov, M. Classical Analysis and ODEs Complex Variables 26C05, 30C15 The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem. |
| title | Maximal univalent disks of real rational functions and Hermite-Biehler polynomials |
| topic | Classical Analysis and ODEs Complex Variables 26C05, 30C15 |
| url | https://arxiv.org/abs/1005.1596 |