Let $ \mathbb{U} $ be the open unit disk, $ \Omega = \{\omega\in \mathbb{C}:{\rm Re}\; \omega > a, -\infty < a < a_{0} < +\infty\} $ with $ a, a_{0}\in\mathbb{R} $, and $ K\geq1 $ and $ K'\geq0 $ as two given constants. In this paper, we study some properties of the class $ \mathcal{H}(\mathbb{U}, \Omega) $ of mappings consisting of those sense-preserving Euclidean harmonic mappings from $ \mathbb{U} $ onto $ \Omega $ with the real, normalized conditions of $ f(0) = a_{0}, f_{z}(0) > 0 $ and $ f_{\overline{z}}(0) $. We first show that a $ (K, K') $-quasiconformal mapping $ f\in\mathcal{H}(\mathbb{U}, \Omega) $ need not be Euclidean Lipschitz continuous. Subsequently, we give a sufficient and necessary condition for $ f\in\mathcal{H}(\mathbb{U}, \Omega) $ to be $ (K, K') $-quasiconformal. Additionally, coefficient estimates, distortion theorems, and area theorems are obtained.
Citation: Deguang Zhong, Meilan Huang. Some properties for $ (K, K') $-quasiconformal harmonic mappings to half-planes[J]. Electronic Research Archive, 2026, 34(4): 2539-2547. doi: 10.3934/era.2026117
Let $ \mathbb{U} $ be the open unit disk, $ \Omega = \{\omega\in \mathbb{C}:{\rm Re}\; \omega > a, -\infty < a < a_{0} < +\infty\} $ with $ a, a_{0}\in\mathbb{R} $, and $ K\geq1 $ and $ K'\geq0 $ as two given constants. In this paper, we study some properties of the class $ \mathcal{H}(\mathbb{U}, \Omega) $ of mappings consisting of those sense-preserving Euclidean harmonic mappings from $ \mathbb{U} $ onto $ \Omega $ with the real, normalized conditions of $ f(0) = a_{0}, f_{z}(0) > 0 $ and $ f_{\overline{z}}(0) $. We first show that a $ (K, K') $-quasiconformal mapping $ f\in\mathcal{H}(\mathbb{U}, \Omega) $ need not be Euclidean Lipschitz continuous. Subsequently, we give a sufficient and necessary condition for $ f\in\mathcal{H}(\mathbb{U}, \Omega) $ to be $ (K, K') $-quasiconformal. Additionally, coefficient estimates, distortion theorems, and area theorems are obtained.
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Deguang Zhong, Meilan Huang. Some properties for $ (K, K') $-quasiconformal harmonic mappings to half-planes[J]. Electronic Research Archive, 2026, 34(4): 2539-2547. doi: 10.3934/era.2026117
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