Related papers: The Schwarz lemma at the boundary

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these…

In this note we establish a Schwarz type inequality for holomorphic mappings between unit balls $B_n$ and $B_m$ in corresponding complex spaces.

This paper deals with a theoretical mathematical analysis of a one-dimensional-moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order $\al$ $\in (0,1)$ is taken in the Caputo's…

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the…

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz…

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…

Based on the recently proved Khavinson conjecture, we establish an inequality of Schwarz-Pick type for harmonic functions on the unit ball of $\mathbb{R}^n$.

We provide an example of an $L^1$ function on the circle, which cannot be the trace of a function of bounded variation of least gradient in the disk. This shows that in theorems on existence and uniqueness of solutions to the least gradient…

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…

For any complex polynomial P having all its zeros in the unit disk, we estimate the rate of change of the argument P (z) when the point z runs through the boundary of this disk.

Indefinite Schwarz-Pick inequalities for holomorphic self-maps of the bidisk are given as application of the spectral theory on Hilbert modules.

Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining…

Let $f$ be a holomorphic, or even meromorphic, function on the unit disc. Plessner's theorem then says that, for almost every boundary point $\zeta $, either (i) $f$ has a finite nontangential limit at $\zeta $, or (ii) the image $f(S)$ of…

In this paper, we consider different versions of the classical Hopf's boundary lemma in the setting of the fractional $p-$Laplacian for $p \geq 2$. We start by providing for a new proof to a Hopf's lemma based on comparison principles.…

Dirac-Schr\"{o}dinger systems play a central role when modeling Dirac bundles and Dirac-Schr\"{o}dinger operators near the boundary, along ends or near other singularities of Riemannian manifolds. In this article we develop the Fredholm…

Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept…

Let $\mathcal{H}$ be the class of all analytic self-maps of the open unit disk $\mathbb{D}$. Denote by $H^n f(z)$ the $n$-th order hyperbolic derivative of $f\in \mathcal H$ at $z\in \mathbb{D}$. For $z_0\in \mathbb{D}$ and $\gamma =…

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the…

We revisit one of Zilber's early results in model-theoretic algebra, viz. definability in Schur's lemma. This takes place in a broader context than the original version from the seventies. The present exposition contains results extracted…

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its…