On the coefficients estimate of K-quasiconformal harmonic mappings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911189862187008 |
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| author | Li, Peijin Ponnusamy, Saminathan |
| author_facet | Li, Peijin Ponnusamy, Saminathan |
| contents | Recently, the Wang et al. \cite{wwrq} proposed a coefficient conjecture for the family ${\mathcal S}_H^0(K)$ of $K$-quasiconformal harmonic mappings $f = h + \overline{g}$ that are sense-preserving and univalent, where $h(z)=z+\sum_{k=2}^{\infty}a_kz^k$ and $g(z)=\sum_{k=1}^{\infty}b_kz^k$ are analytic in the unit disk $|z|<1$, and the dilatation $ω=g'/h'$ satisfies the condition $|ω(z)| \leq k<1$ for $\ID$, with $K=\frac{1+k}{1-k}\geq 1$. The main aim of this article is provide an affirmative answer in support of this conjecture by proving this conjecture for every starlike function (resp. close-to-convex function) from $\mathcal{S}^0_H(K)$. In addition, we verify this conjecture also for typically real $K$-quasiconformal harmonic mappings. Also, we establish sharp coefficients estimate of convex $K$-quasiconformal harmonic mappings. By doing so, our work provides a document in support of the main conjecture of Wang et al.. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_08284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the coefficients estimate of K-quasiconformal harmonic mappings Li, Peijin Ponnusamy, Saminathan Complex Variables Primary 31A05, Secondary 30C55, 30C62 Recently, the Wang et al. \cite{wwrq} proposed a coefficient conjecture for the family ${\mathcal S}_H^0(K)$ of $K$-quasiconformal harmonic mappings $f = h + \overline{g}$ that are sense-preserving and univalent, where $h(z)=z+\sum_{k=2}^{\infty}a_kz^k$ and $g(z)=\sum_{k=1}^{\infty}b_kz^k$ are analytic in the unit disk $|z|<1$, and the dilatation $ω=g'/h'$ satisfies the condition $|ω(z)| \leq k<1$ for $\ID$, with $K=\frac{1+k}{1-k}\geq 1$. The main aim of this article is provide an affirmative answer in support of this conjecture by proving this conjecture for every starlike function (resp. close-to-convex function) from $\mathcal{S}^0_H(K)$. In addition, we verify this conjecture also for typically real $K$-quasiconformal harmonic mappings. Also, we establish sharp coefficients estimate of convex $K$-quasiconformal harmonic mappings. By doing so, our work provides a document in support of the main conjecture of Wang et al.. |
| title | On the coefficients estimate of K-quasiconformal harmonic mappings |
| topic | Complex Variables Primary 31A05, Secondary 30C55, 30C62 |
| url | https://arxiv.org/abs/2504.08284 |