On Zalcman's and Bieberbach conjectures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917205693693952 |
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| author | Krushkal, Samuel L. |
| author_facet | Krushkal, Samuel L. |
| contents | The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n > 2$, with equality only for the Koebe function and its rotations. The conjecture was proved by the author for $n \le 6$ (using geometric arguments related to the Ahlfors-Schwarz lemma) and remains open for $n \ge 7$.
The main theorem of this paper states that these conjectures are equivalent and provides their simultaneous proof for all $n \ge 3$ combining the indicated geometric arguments with a new author's approach to extremal problems for holomorphic functions based on lifting the rotationally homogeneous coefficient functionals to the Bers fiber space over universal Teichmuller space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10584 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Zalcman's and Bieberbach conjectures Krushkal, Samuel L. Complex Variables The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n > 2$, with equality only for the Koebe function and its rotations. The conjecture was proved by the author for $n \le 6$ (using geometric arguments related to the Ahlfors-Schwarz lemma) and remains open for $n \ge 7$. The main theorem of this paper states that these conjectures are equivalent and provides their simultaneous proof for all $n \ge 3$ combining the indicated geometric arguments with a new author's approach to extremal problems for holomorphic functions based on lifting the rotationally homogeneous coefficient functionals to the Bers fiber space over universal Teichmuller space. |
| title | On Zalcman's and Bieberbach conjectures |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2601.10584 |