Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma

Abstract

In this paper, we first obtain a Schwarz-Pick type lemma for the holomorphic self-mapping of the unit disk with respect to the $ q $-distance. Second, we establish the general Schwarz-Pick lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality. As an application, it is proven that this mapping is Lipschitz continuous with respect to the $ q $-distance under certain conditions. Moreover, the corresponding explicit Lipschitz constant is given. Third, it is proved that there exists a self-mapping of the unit disk satisfying the Poisson differential inequality, which does not meet conditions of the boundary Schwarz lemma. Finally, with some additional conditions, a boundary Schwarz lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality is established.

Keywords

• schwarz-pick lemma
• poisson differential inequality
• $ q $-distance
• boundary schwarz lemma
• explicit constant