Computational Methods and Function Theory最新文献

In this article, we present the symmetry group of a global slice Dirac operator and its iterated ones. Further, the explicit forms of intertwining operators of the iterated global slice Dirac operator are given. At the end, we introduce a variant of the global slice Dirac operator, which allows functions considered to be defined on the whole Euclidean space. The invariance property and the intertwining operators of this variant of the global slice Dirac operator are also presented.

Let f be an entire function and L(f) a linear differential polynomial in f with constant coefficients. Suppose that f, (f'), and L(f) share a meromorphic function (alpha (z)) that is a small function with respect to f. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function (alpha ) must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then f can be obtained from each solution. Examples suggest that only rarely do single-valued solutions (alpha (z)) exist, and even then they are not always small functions for f.

A boundary integral equation method is presented for fast computation of the analytic capacities of compact sets in the complex plane. The method is based on using the Kerzman–Stein integral equation to compute the Szegő kernel and then the value of the derivative of the Ahlfors map at the point at infinity. The proposed method can be used for domains with smooth and piecewise smooth boundaries. When combined with conformal mappings, the method can be used for compact slit sets. Several numerical examples are presented to demonstrate the efficiency of the proposed method. We recover some known exact results and corroborate the conjectural subadditivity property of analytic capacity.

In this paper, we consider the second order linear differential equation

where A, B and F with (Bnot equiv 0) are entire functions. We find some appropriate conditions on A, B and F in terms of the (varphi )-order which guarantee that every non-constant entire solution f of (†) has infinite (varphi )-order, along with an additional relation between the hyper-(varphi )-order of f and the (varphi )-order of the dominating coefficient in (†).

In this paper, we introduce a new metric (tilde{c}) which is associated with the domain boundary for a Ptolemy space (X, d). Moreover, we study inclusion relations of (tilde{c}) metric balls and some related hyperbolic type metric balls in subdomains of ({mathbb {R}}^n.) In addition, we study distortion properties of the (tilde{c}) metric under Möbius transformations of the unit ball and quasiconformality of bilipschitz mappings in the (tilde{c}) metric.

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces (A^p_w) with arbitrary (non-negative and integrable) radial weights w in the case (1le p<infty ). We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption (liminf _{rrightarrow 0^+} w(r)>0), we show that the principle fails whenever (0<p<1).

In this paper, for (p>1) and (s>1), we characterize completely the boundedness and compactness of a Cesàro-like operator from the Besov space (B_p) into a Banach space X between the mean Lipschitz space (Lambda ^s_{1/s}) and the Bloch space. In particular, for (p=s=2), we complete a previous result from the literature.

We prove that the metric completion of the intrinsic length space associated with a simply and rectifiably connected plane set is a Hadamard space. We also characterize when such a space is Gromov hyperbolic.

In this note, we prove a Paley–Wiener Theorem for the Mehler–Fock transform. In particular, we show that it induces an isometric isomorphism from the Hardy space (mathcal H^2(mathbb C^+)) onto (L^2(mathbb R^+,( 2 pi )^{-1} t sinh (pi t) , dt ) ). The proof we provide here is very simple and is based on an old idea that seems to be due to G. R. Hardy. As a consequence of this Paley–Wiener theorem we also prove a Parseval’s theorem. In the course of the proof, we find a formula for the Mehler–Fock transform of some particular functions.

Due to the limitations of the harmonic convolution defined by Clunie and Sheil Small (Ann Acad Sci Fenn Ser A I Math 9:3–25, 1984), a new product (otimes ) has been recently introduced (2021) for two harmonic functions defined in an open unit disk of the complex plane. In this paper, the radius of univalence (and other radii constants) for the products (Kotimes K) and (Lotimes f) are computed, where K denotes the harmonic Koebe function, L denotes the harmonic right half-plane mapping and f is a sense-preserving harmonic function defined in the unit disk with certain constraints. In addition, several conditions on harmonic function f are investigated under which the product (Lotimes f) is sense-preserving and univalent in the unit disk.