Computational Methods and Function Theory最新文献

Let (Gsubsetneq {mathbb {R}}^n) be a domain, where (nge 2). Let (k_G) and (j_G) be the quasihyperbolic metric and the distance ratio metric on G, respectively. In the present paper, we prove that the identity map of ((G,k_G)) onto ((G,j_G)) is quasisymmetric if and only if it is bilipschitz. To classify domains of ({mathbb {R}}^n) into various types according to the behaviors of their quasihyperbolic metrics, we define a uniformity exponent for every proper subdomain of ({mathbb {R}}^n) and prove that this exponent may assume any value in ({0}cup [1,infty ]). Moreover, we study the properties of domains of uniformity exponent 1 and show by an example that such a domain may be neither quasiconvex nor accessible.

If (mu ) is a positive Borel measure on the interval [0, 1) we let ({mathcal {H}}_mu ) be the Hankel matrix ({mathcal {H}}_mu =(mu _{n, k})_{n,kge 0}) with entries (mu _{n, k}=mu _{n+k}), where, for (n,=,0, 1, 2, ldots ), (mu _n) denotes the moment of order n of (mu ). This matrix formally induces an operator, called also ({mathcal {H}}_mu ), on the space of all analytic functions in the unit disc ({mathbb {D}}) as follows: If f is an analytic function in ({mathbb {D}}), (f(z)=sum _{k=0}^infty a_kz^k), (zin {{mathbb {D}}}), ({mathcal {H}}_mu (f)) is formally defined by

This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators (H_mu ) acting on the Bergman spaces (A^p), (1le p<infty ). Among other results, we give a complete characterization of those (mu ) for which ({mathcal {H}}_mu ) is bounded or compact on the space (A^p) when p is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of (mathcal H_mu ) on (A^p) for the other values of p, as well as on its membership in the Schatten classes ({mathcal {S}}_p(A^2)).

The purpose of this paper is to establish new characterizations of concave functions f defined in ({mathbb {D}}) in terms of the operator (1+zf''/f'), the Schwarzian derivative and the lower order. We will distinguish the cases when the omitted set is bounded or unbounded, and in the latter case, we will address the subclasses determined by the angle at infinity.

We employ tools from complex analysis to construct the (*)-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the (*)-exponential; we establish sufficient conditions for the (*)-product of two (*)-exponentials to also be a (*)-exponential; we calculate the slice derivative of the (*)-exponential of a regular function.

Inspired by the questions Gundersen and Yang proposed, we investigate the exact forms of the entire solutions of the following two types of binomial differential equations

where a, b, c are polynomials with no common zeros satisfying (abcnot equiv 0), and q is a non-constant polynomial.

We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, (nge 3). The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if ( { f_{n} }_{n=1}^{infty } ) is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion (H({f_{n}}))), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence ( {f_{n} }_{n=1}^{infty } ) with ( {f_{n}}rightarrow {f}) locally uniformly and with (limsup _{nrightarrow infty } H( {f_{n}})<H( {f})). Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each (alpha <sqrt{2}) there is ({f_{n}}rightarrow {f}) locally uniformly with f affine and

We conjecture (sqrt{2}) to be best possible.

We study the linear dilatation of the mappings satisfying an inverse Poletsky inequality in metric spaces. We also show that under certain conditions such mappings are quasiregular.

We provide an elementary derivation of the Bessel analog of the celebrated Riesz composition formula and use the former to effortlessly derive the latter.

Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.

We show that the Poincaré inequality holds on an open set (Dsubset mathbb {R}^n) if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict inradius of D measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet–Laplacian.