Abstract
Let \({\alpha },{\beta }\in (-1,\infty )\) such that \({\alpha }+{\beta }>-1\). Given two continuous functions \(g \in \mathcal {C}(\overline{{\mathbb D}})\) and \(f\in \mathcal {C}({\mathbb T})\), we provide various Schwarz type lemmas for mappings u satisfying the inhomogeneous \(({\alpha },{\beta })\)-harmonic equation \(L_{{\alpha },{\beta }}u=g\) in \({\mathbb D}\) and \(u=f\) in \({\mathbb T}\), where \({\mathbb D}\) is the unit disc of the complex plane \({\mathbb C}\) and \({\mathbb T}=\partial {\mathbb D}\) is the unit circle. The obtained results provide a significant improvement over previous research on the subject.
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The authors would like to thank the referees for insightful comments which led to significant improvements in the paper.
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Communicated by Rosihan M. Ali.
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Khalfallah, A., Mhamdi, M. Schwarz Type Lemmas for Generalized Harmonic Functions. Bull. Malays. Math. Sci. Soc. 47, 53 (2024). https://doi.org/10.1007/s40840-023-01646-4
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DOI: https://doi.org/10.1007/s40840-023-01646-4
Keywords
- Schwarz lemma
- \(({\alpha },{\beta })\)-Harmonic mapping
- Schwarz–Pick lemma
- Weighted Green function
- \(({\alpha },{\beta })\)-Harmonic equation