Schwarz Type Lemmas for Generalized Harmonic Functions

Let \({\alpha },{\beta }\in (-1,\infty )\) $\alpha ,\beta \in (-1,\infty )$ such that \({\alpha }+{\beta }>-1\) $\alpha +\beta >-1$ . Given two continuous functions \(g \in \mathcal {C}(\overline{{\mathbb D}})\) $g\in \mathcal{C}(\overline{\mathbb{D}})$ and \(f\in \mathcal {C}({\mathbb T})\) $f\in \mathcal{C}(\mathbb{T})$ , we provide various Schwarz type lemmas for mappings u satisfying the inhomogeneous \(({\alpha },{\beta })\) $(\alpha ,\beta )$ -harmonic equation \(L_{{\alpha },{\beta }}u=g\) $L_{\alpha ,\beta }u=g$ in \({\mathbb D}\) $\mathbb{D}$ and \(u=f\) $u=f$ in \({\mathbb T}\) $\mathbb{T}$ , where \({\mathbb D}\) $\mathbb{D}$ is the unit disc of the complex plane \({\mathbb C}\) $\mathbb{C}$ and \({\mathbb T}=\partial {\mathbb D}\) $\mathbb{T}=\partial \mathbb{D}$ is the unit circle. The obtained results provide a significant improvement over previous research on the subject.