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The Schwarz-Pick lemma for slice regular functions  [PDF]
Cinzia Bisi,Caterina Stoppato
Mathematics , 2012, DOI: 10.1512/iumj.2012.61.5076
Abstract: The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.
The Schwarz-Pick lemma of high order in several variables  [PDF]
Shaoyu Dai,Huaihui Chen,Yifei Pan
Mathematics , 2011,
Abstract: We prove a high order Schwarz-Pick lemma for mappings between unit balls in complex spaces in terms of the Bergman metric. From this lemma, Schwarz-Pick estimates for partial derivatives of arbitrary order of mappings are deduced.
A Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces  [PDF]
Shaoyu Dai,Yifei Pan
Mathematics , 2013,
Abstract: In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces. This extends the classical Schwarz-Pick lemma and the related result proved by Pavlovic.
The Schwarz lemma at the boundary  [PDF]
Steven G. Krantz
Mathematics , 2010,
Abstract: The most classical version of the Schwarz lemma involves the behavior at the origin of a bounded, holomorphic function on the disc. Pick's version of the Schwarz lemma allows one to move the origin to other points of the disc. In the present paper we explore versions of the Schwarz lemma at a boundary point of a domain (not just the disc). Estimates on derivatives of the function, and other types of estimates as well, are considered. We review recent results of several authors, and present some new theorems as well.
A new variant of the Schwarz-Pick-Ahlfors lemma  [PDF]
Robert Osserman
Mathematics , 1998,
Abstract: We prove a ``general shrinking lemma'' that resembles the Schwarz--Pick--Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that {\em all} distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz--Pick--Ahlfors-type lemmas to the comparison theorems of Riemannian geometry.
A note on Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n  [PDF]
Dai Shaoyu,Pan Yifei
Mathematics , 2013,
Abstract: In this paper we prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n.
A Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball  [PDF]
Shaoyu Dai,Yifei Pan
Mathematics , 2013,
Abstract: In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.
Linear connectivity, Schwarz-Pick lemma and univalency criteria for planar harmonic mappings  [PDF]
Shaolin Chen,Saminathan Ponnusamy,Antti Rasila,Xiantao Wang
Mathematics , 2014,
Abstract: In this paper, we first establish the Schwarz-Pick lemma of higher-order and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
A reverse Schwarz--Pick inequality  [PDF]
Konstantin M. Dyakonov
Mathematics , 2013, DOI: 10.1007/s40315-013-0029-8
Abstract: We prove a kind of "reverse Schwarz--Pick lemma" for holomorphic self-maps of the disk. The result becomes especially clear-cut for inner functions and casts new light on their derivatives.
Weighted Lipschitz continuity, Schwarz-Pick's Lemma and Landau-Bloch's theorem for hyperbolic-harmonic mappings in $\mathbb{C}^{n}$  [PDF]
Sh. Chen,S. Ponnusamy,X. Wang
Mathematics , 2012,
Abstract: In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $\alpha$-Bloch spaces.
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