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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.01592 |
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| _version_ | 1866918329492439040 |
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| author | Hernández, Rodrigo Martín, María J. Pérez-González, Fernando Wołoszkiewicz-Cyll, Magdalena |
| author_facet | Hernández, Rodrigo Martín, María J. Pérez-González, Fernando Wołoszkiewicz-Cyll, Magdalena |
| contents | We determine completely the analytic functions $φ$ in the unit disk $\mathbb D$ such that for all (normalized) orientation-preserving harmonic mappings $f=h+\overline g$ produced by the shear construction with $h+g=φ$, the condition that each $f$ maps $\mathbb D$ onto a convex domain holds. As a consequence, we obtain the following more general result: for a given complex number $η$, with $|η|=1$, we characterize those holomorphic mappings $φ$ in $\mathbb D$ such that every harmonic function $f=h+\overline g$ as above with $h-ηg=φ$ maps $\mathbb D$ onto a convex domain. The resulting functions are mappings onto a half-plane and mappings onto a strip, and the shear direction, determined by the parameter $η$ above, is parallel to the linear boundaries of the half-planes and strips. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01592 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Always-convex harmonic shears Hernández, Rodrigo Martín, María J. Pérez-González, Fernando Wołoszkiewicz-Cyll, Magdalena Complex Variables 31A05, 30C45 We determine completely the analytic functions $φ$ in the unit disk $\mathbb D$ such that for all (normalized) orientation-preserving harmonic mappings $f=h+\overline g$ produced by the shear construction with $h+g=φ$, the condition that each $f$ maps $\mathbb D$ onto a convex domain holds. As a consequence, we obtain the following more general result: for a given complex number $η$, with $|η|=1$, we characterize those holomorphic mappings $φ$ in $\mathbb D$ such that every harmonic function $f=h+\overline g$ as above with $h-ηg=φ$ maps $\mathbb D$ onto a convex domain. The resulting functions are mappings onto a half-plane and mappings onto a strip, and the shear direction, determined by the parameter $η$ above, is parallel to the linear boundaries of the half-planes and strips. |
| title | Always-convex harmonic shears |
| topic | Complex Variables 31A05, 30C45 |
| url | https://arxiv.org/abs/2507.01592 |