Abstract
In this paper, we establish some Schwarz type lemmas for map**s \(\Phi \) satisfying the inhomogeneous biharmonic Dirichlet problem \( \Delta (\Delta (\Phi )) = g\) in \({\mathbb D}\), \(\Phi =f\) on \({\mathbb T}\) and \(\partial _n \Phi =h\) on \({\mathbb T}\), where g is a continuous function on \(\overline{{\mathbb D}}\), f, h are continuous functions on \({\mathbb T}\), where \({\mathbb D}\) is the unit disc of the complex plane \({\mathbb C}\) and \({\mathbb T}=\partial {\mathbb D}\) is the unit circle. To reach our aim, we start by investigating some properties of generalized harmonic functions called \(T_\alpha \)-harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when \(\alpha >0\).
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We would like to thank the referee for insightful comments which led to significant improvements in the paper.
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Communicated by Adrian Constantin.
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Khalfallah, A., Haggui, F. & Mhamdi, M. Generalized harmonic functions and Schwarz lemma for biharmonic map**s. Monatsh Math 196, 823–849 (2021). https://doi.org/10.1007/s00605-021-01619-4
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DOI: https://doi.org/10.1007/s00605-021-01619-4
Keywords
- Schwarz’s lemma
- Boundary Schwarz’s lemma
- Landau theorem
- Biharmonic equations
- \(T_\alpha \)-harmonic map**s