Schwarz type lemmas and their applications in Banach spaces

Abstract

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls of C n and to obtain some sharp boundary Schwarz type lemmas for holomorphic mappings in Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Banach spaces. Additionally, using a relatively simple method of proof, we prove some sharp Schwarz-Pick type estimates of pluriharmonic mappings in JB * -triples, and the obtained results provide the improvements and generalizations of the corresponding results in [9].

The Boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalisations

“…After writing the final version of the manuscript [19] Hamada turned attention 1 to the arxiv paper [7]. In [7], the authors generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms for Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Banach spaces. Recently, Hamada and Kohr published paper [12], where authors discussed rigidity theorems on the boundary for holomorphic mappings.…”

“…In the Section 2 we will improve this estimate. Next S. Chen, H. Hamada, S. Ponnusamy, R. Vijayakumar in [7] observed that using [ [11], Proposition 1.8] and the arguments similar to those in the proof of their Theorem 3.3 [7] one can obtain a better estimate:…”

“…Theorem 1.3 (Theorem 3.3 [7]). Suppose that and are the unit balls of the complex Banach spaces and , respectively, and ∶ → is a pluriharmonic mapping.…”

“…We will use notation from [7]. Let and be real or complex Banach spaces with norm | ⋅ | and | ⋅ | respectively.…”

“…Although it is one of the simplest results showing the rigidity of holomorphic functions Schwarz lemma has been generalized in various directions and it has become a crucial theme in many branches of research in Mathematics for more than a hundred years to the present day. There is vast literature related to the subject, but here we cite mainly recent papers; for a more complete list of references see [1,7,20,21] and the references therein for more fundamental results.…”