Related papers: A reverse Schwarz--Pick inequality
This paper establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions defined on the complex unit ball $\mathbb B^n$. The proof of this main result simultaneously provides a solution to a natural extension…
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.…
In this note we establish a Schwarz type inequality for holomorphic mappings between unit balls $B_n$ and $B_m$ in corresponding complex spaces.
Two different problems are considered here. First, a characterization of sampling sequences for the class of analytic functions from the disc into itself is given. Second, a version of Schwarz-Pick Lemma for $n$ points leads to an…
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…
We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we…
We establish some estimates of the the angular derivatives from below for holomorphic self-maps of the unit disk at one and two fixed points of the unit circle provided there is no fixed point inside the unit disk. The results complement…
We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.
It is well-known that the classical Schwarz lemma yields an explicit comparison of two Hermitian metrics with uniform constant negative curvature bounds through holomorphic maps between complex manifolds. In this paper, we establish Schwarz…
The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $\alpha$-harmonic mapping $u=P_{\alpha}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the…
We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and…
We consider the multi-point Schwarz-Pick lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point…
Suppose $w$ is a sense-preserving harmonic mapping of the unit disk $\mathbb{D}$ such that $w(\mathbb{D})\subseteq\mathbb{D}$ and $w$ has a zero of order $p\geq1$ at $z=0$. In this paper, we first improve the Schwarz lemma for $w$, and…
There is a known generalization of the classical Schwarz lemma to holomorphic functions from the polydisk to the disk. In this paper, we characterize those functions which satisfy equality everywhere in this generalized inequality: they are…
The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=\gamma_0$ and…
In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Carath\'eodory theorem for univalent…
A sharp version of a recent inequality of Kovalev and Yang on the ratio of the $(H^1)^\ast$ and $H^4$ norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the…
In this paper, we prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary…
In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved…
We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L.Harris via the generation theory for…