Related papers: The Schwarz Lemma: An Odyssey
The purpose of this note is to discuss the real analogue of the Schwarz lemma from complex analysis.
The aim of this article is to give an elementary proof of the fact that the Schwarz-Pick Lemma follows from the Ahlfors-Schwarz-Pick Lemma.
This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and the applications.
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…
The main purpose of this paper is to establish a Schwarz lemma for the solutions to the Dirichlet problems for the invariant Laplacians. The obtained result of this paper is a generalization of the corresponding known results [11, Theorem…
We establish some Schwarz type Lemmas for mappings defined on the unit disk with bounded Laplacian. Then we apply these results to obtain boundary versions of the Schwarz lemma.
We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The…
Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.
We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area and total spherical curvature. These results can…
Notes to lectures on the epsilon calculus, covering axioms, semantics, completeness, and the first epsilon theorem.
The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and…
For $n\geq3$, $m\geq1$ and a given continuous function $g:~\Omega\rightarrow\mathbb{R}^{m}$, we establish some Schwarz type lemmas for mappings $f$ of $\Omega$ into $\mathbb{R}^{m}$ satisfying the PDE: $\Delta f=g$, where $\Omega$ is a…
A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to…
In this paper, we establish a boundary Schwarz lemma for solutions to non-homogeneous biharmonic equations.
This is a writeup of a talk given at the Oskar Klein Centenery Symposium, Stockholm, September 19-21, 1994. It is an essay on the black hole information paradox and its connection with thermodynamics and the foundations of quantum…
This is an appendix to our paper "An update of the Hirsch Conjecture" (arXiv:0907.1186), containing proofs of some of the results and comments that were omitted in it.
We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of `nearly unit' vectors.
These are lecture notes that are based on the lectures from a class I taught on the topic of Randomized Linear Algebra (RLA) at UC Berkeley during the Fall 2013 semester.
For a given continuous function $g:~\Omega\rightarrow\mathbb{C}$, we establish some Schwarz type Lemmas for mappings $f$ in $\Omega$ satisfying the {\rm PDE}: $\Delta f=g$, where $\Omega$ is a subset of the complex plane $\mathbb{C}$. Then…
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to…