Related papers: Sharp Schwarz-type lemmas for the spectral unit ba…
The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions…
We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and Kalaj (F.~Forstneri{\v{c}} and D.~Kalaj. \newblock…
We prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.
We obtain upper bounds for the norm of the Schwarzian derivative of convex holomorphic mappings defined on the polydisk and the unit ball in $\mathbb{C}^n$. For coordinate-wise convex mappings on the polydisk, we derive a sharp estimate…
In this paper we establish several invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphic selfmaps of strongly convex domains in $\mathbb C^N$ in the spirit of…
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…
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville…
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.
In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic…
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…
In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the…
We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in $\MM_n(\CC^n)$.
In this paper, we obtain a Schwarz lemma for holomorphic mappings from the unit polydisc $P_m$ into the unit polydisc $P_n$, here $P_m$ and $P_n$ are endowed with $\mbox{Aut}(P_m)$-invariant K\"ahelr-Berwald metric $F_{t,k}$ and…
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In…
We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…
In this paper we prove a Schwarz-Pick lemma for bounded complex-valued harmonic functions in the unit ball of R^n.
Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new…
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to…
Let $\alpha>-1$ and assume that $f$ is $\alpha-$harmonic mapping defined in the unit disk that belongs to the Hardy class $h^p$ with $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|) \|f^\ast\|_p$ and $|Df(z)|\le…
We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension…