arXiv subjects
Publications and source records attributed to Matti Vuorinen.
In this paper, we continue our investigation of function spaces on certain classes of complex-valued functions. In particular, we give characterizations on Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we present applications of our results to certain nonlinear PDEs.
Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count.
In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Turán type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Turán type inequality for a series considered by Ramanujan, involving the digamma function.
We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of Möbius transformations of the unit ball in ${\mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.
A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $G\subsetneq\Rn$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $\Bn$ or the upper half space $\Hn$. We also obtain the sharp Lipschitz constant for a Möbius transformation $f: G\rightarrow G'$ between domains $G$ and $G'$ in $\Rn$ with respect to the metrics $v_G$ and $v_{G'}$. For instance, in the case $G=G'=\Bn$ the result is sharp.
Given a pair of points in the hyperbolic half plane or the unit disk, we provide a simple construction of the midpoint of the hyperbolic geodesic segment joining the points.
We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings from the unit disk onto itself and obtain sharp results in this case, too.
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ 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 Väisälä in the affirmative.
We give study the Lipschitz continuity of Möbius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
An improved version of quasiinvariance property of the quasihyperbolic metric under Möbius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is proved. Finally, several inequalities between the quasihyperbolic metric and other commonly used metrics such as the hyperbolic metric of the unit ball and the chordal metric are established.
In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the properties of monotonicity and convexity of this function. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.
Teichmüller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain $D \subset {\mathbb R}^n\,,n\ge 2\,,$ we consider the class of all $K$- quasiconformal maps of $D$ onto itself with identity boundary values and Teichmüller's problem in this context. Given a map $f$ of this class and a point $x\in D\,,$ we show that the maximal dilatation of $f$ has a lower bound in terms of the distance of $x$ and $f(x)$ in the distance ratio metric. For instance, convex domains, bounded domains and domains with uniformly perfect boundaries are studied.
The authors survey recent results in special functions of classical analysis and geometric function theory, in particular the circular and hyperbolic functions, the gamma function, the elliptic integrals, the Gaussian hypergeometric function, power series, and mean values.
Inclusion relations of metric balls defined by the hyperbolic, the quasihyperbolic, the $j$-metric and the chordal metric will be studied. The hyperbolic metric, the quasihyperbolic metric and the $j$-metric are considered in the unit ball.
For a given domain $D$ in the extended complex plane $\bar{\mathbb C}$ with an accessible boundary point $z_0 \in \partial D$ and for a subset $E \subset {D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\rc E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $r(D\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0.$ Here $r(G,z)$ denotes the inner radius at $z$ of the connected component of the set $G$ containing the point $z.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when the domain $D$ is the upper half plane and $z_0=\infty$ the capacity $\rc E$ coincides with the well-known half-plane capacity ${\hc} E.$ Some properties of the relative capacity are proven, including the behavior of this capacity under various forms of symmetrization and under some other geometric transformations. Some applications to bounded holomorphic functions of the unit disk are given.
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 original 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.
We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the Dirichlet-Neumann problem for the Laplace equation. Several experimental results, with error estimates, are reported. Our main method makes use of an $hp$-FEM algorithm, which enables computations in the case of complicated geometry. For simple geometries, good agreement with computational results based on the SC Toolbox, is observed. We also use the reciprocal error estimation method introduced in our earlier paper to validate our numerical results. In particular, exponential convergence, in accordance with the theory of Babu\vska and Guo, is demonstrated.
The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Lazarević's inequality, Huygens-type inequalities, Wilker-type inequalities, and Cuza-Huygens-type inequalities, are generalized to the case of generalized functions.