Abstract
The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: \(\Delta (\Delta f)=g\), where g : \(\overline{{\mathbb D}}\rightarrow {\mathbb {C}}\) is a continuous function and \(\overline{{\mathbb D}}\) denotes the closure of the unit disk \({\mathbb D}\) in the complex plane \({\mathbb {C}}\). In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality
where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.
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Acknowledgements
The research was partly supported by the Natural Science Foundation of China (No. 2022JJ40112), Scientific Fund of Hunan Provincial Education Department (No. 20B118), and NSF of Hebei Science (No. 2021201006).
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Li, P., Li, Y., Luo, Q. et al. On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations. Mediterr. J. Math. 20, 142 (2023). https://doi.org/10.1007/s00009-023-02344-y
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DOI: https://doi.org/10.1007/s00009-023-02344-y