Dual Smale's mean value conjecture for odd polynomials
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915563224170496 |
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| author | Tang, Quanyu |
| author_facet | Tang, Quanyu |
| contents | We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $ζ$ of $P$ such that $$ \left|\frac{P(ζ)}ζ\right| \ge \frac1d. $$This result can be regarded as a dual counterpart of T. W. Ng's theorem on Smale's mean value conjecture for odd polynomials with nonzero linear term [J. Aust. Math. Soc. 75 (2003), 409--411]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dual Smale's mean value conjecture for odd polynomials Tang, Quanyu Complex Variables Primary 30C10 We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $ζ$ of $P$ such that $$ \left|\frac{P(ζ)}ζ\right| \ge \frac1d. $$This result can be regarded as a dual counterpart of T. W. Ng's theorem on Smale's mean value conjecture for odd polynomials with nonzero linear term [J. Aust. Math. Soc. 75 (2003), 409--411]. |
| title | Dual Smale's mean value conjecture for odd polynomials |
| topic | Complex Variables Primary 30C10 |
| url | https://arxiv.org/abs/2510.16875 |