We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.
We give study the Lipschitz continuity of Mobius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
Given a domain $G subsetneq Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D subset G$. In each case, distances in the subdomain are always larger than in the o
riginal domain. Our goal is to show that, in several cases, one can prove a stronger domain monotonicity statement. We also show that under special hypotheses we have inequalities in the opposite direction.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2 and that $Dvarsubsetneq E$ and $Dvarsubsetneq E$ are uniform domains with homogeneously dense boundaries. We consider the class of all $varphi$-FQC (freely $varphi$-quasicon
formal) maps of $D$ onto $D$ with bilipschitz boundary values. We show that the maps of this class are $eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Holder condition. Moreover, replacing the class $varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C,,$ depending only on the function $varphi,,$ such that for all $xin D$, the quasihyperbolic distance satisfies $k_D(x,f(x))leq C$.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2, that $Dsubset E$ and $Dsubset E$ are domains, and that $f: Dto D$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform
domains: Suppose that $f$ is a freely quasiconformal mapping and that $D$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Vaisala in the affirmative.
Numpy and SciPy are program libraries for the Python scripting language, which apply to a large spectrum of numerical and scientific computing tasks. The Sage project provides a multiplatform software environment which enables one to use, in a unifie
d way, a large number of software components, including Numpy and Scipy, and which has Python as its command language. We review several examples, typical for scientific computation courses, and their solution using these tools in the Sage environment.
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
Let $E$ be a continuum in the closed unit disk $|z|le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $nge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_
k$ on a prescribed circle $|z| = rho, 0 <rho <1, k=1,...,n,. $ It is shown that for some increasing function $Psi,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures $$ Psi^{-1}[ frac{1}{n} sum_{k=1}^{k} Psi(omega(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $omega(rho, E^*, D^*),,$ where $E^* = {z: z^n in [-1,0] }$ and $D^* ={z: |z|<1, |{rm arg} z| < pi/n} ,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $inf_{E} max_{k=1,...,n} omega(a_k,E, D_k),$ for arbitrary points of the circle $|z| = rho ,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = rho ,.$
