The Boundary Schwarz Lemma for Harmonic and Pluriharmonic Map**s and Some Generalizations

Abstract

We use the improvement of the classical Schwarz lemmas for planar harmonic map**s into the sharp form, in order to provide some applications to sharp boundary Schwarz type lemmas for holomorphic and in particular pluriharmonic map**s between the unit balls in Hilbert and Banach spaces. In the second part of this article, using Burget’s estimate we establish the sharp boundary Schwarz type lemmas for harmonic map**s between finite dimensional balls. Since here we do not suppose in general that maps fix the origin this is a generalization of the result, previously established by David Kalaj, for harmonic functions. At the end of this section, we derived some interesting conclusion considering hyperbolic-harmonic functions in the unit ball, which shows that Hopf’s lemma is not applicable for those functions.

5. Appendix

Motivated by the role of the Schwarz lemma in Complex Analysis and numerous fundamental results, see for instance [4, 19] and references therein, in 2016, the first author [2] has posted on ResearchGate the project “Schwarz lemma, the Carathéodory and Kobayashi Metrics and Applications in Complex Analysis”.⁴

In this project and in [4], cf. also [21], we developed the method related to holomorphic map**s with strip codomain (we refer to this method as the approach via the Schwarz–Pick lemma for holomorphic maps from the unit disc into a strip; shortly “planar strip method”). It is worth mentioning that the Schwarz lemma has been generalized in various directions; see [2, 13, 14].

In joint paper of the first author with M. Svetlik [13] using “planar strip method” which is a completely different approach than B. Burgeth [9], we get a simple proof of an optimal version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map), which improves H. W. Hethcote result⁵.

In joint paper of the first author with A. Khalfallah and M. Mhamdi [12], some properties of map**s admitting a Poisson-type integral representations and the boundary Schwarz lemma were considered.

Presently on this project the first author works with some of his associates: A. Khalfallah, M. Arsenović, M. Svetlik, M. Mhamdi, B. Purtić, H.P. Li, J. Gajić and the second author of this paper.

Chinese mathematicians have made a great contribution to this field but here we will mention only some whose results are related to our results. For some interesting complex n-dimensional generalisations of classical Schwarz lemma type results see Jian-Feng Zhu’s articles [22] and [23]. In paper [24] the authors proved Schwarz lemma on the boundary for holomorphic map**s between unit balls in ${\mathbb {C}}^n$, and some of theirs rigidity properties. Generalization of this theorem, for separable complex Hilbert space was given by Z. Chen, Y. Liu and Y. Pan in [15]. While proving Proposition 2.9, we independently proved Lemma 2.7, but later found that result proven in [15], as it can be seen in corresponding reference. In [25] the authors proved a higher order Schwarz-Pick lemma for holomorphic map**s between unit balls in complex Hilbert spaces.

For generalizations of Schwarz lemmas for planar harmonic map**s into the sharp forms of Banach spaces we refer the interested reader to Chen, Hamada et al. [26] (cf. also [6]).