Computational Methods and Function Theory最新文献

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.

Let h be a sense-preserving homeomorphism of the unit circle ({mathbb {S}}) and (Phi (h)) the Douady–Earle extension of h to the closure of the open disk ({mathbb {D}}). In this paper, assuming that h is differentiable at a point (xi in {mathbb {S}}) with (alpha )-Hölder convergence rate for some (0<alpha <1), we prove a similar regularity for (Phi (h)) near (xi ) on ({mathbb {D}}) in any non-tangential direction towards (xi ).

In this paper, we prove the existence and uniqueness of the solution f(z, t) of the Loewner PDE with normalization (Df(0,t)=e^{tA}), where (Ain L(X,X)) is such that (k_+(A)<2m(A)), on the unit ball of a separable reflexive complex Banach space X. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings v(z, s, t) with normalization (Dv(0,s,t)=e^{-(t-s)A}) for (tge sge 0), where (m(A)>0), which satisfy the semigroup property on the unit ball of a complex Banach space X. We further obtain the biholomorphicity of A-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space X. We prove the existence of the biholomorphic solutions f(z, t) of the Loewner PDE with normalization (Df(0,t)=e^{tA}) on the unit ball of a separable reflexive complex Banach space X. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013.

Let S(p) be the class of all meromorphic univalent functions defined in the unit disc ({mathbb D}) of the complex plane with a simple pole at (z=p) and normalized by the conditions (f(0)=0) and (f'(0)=1). In this article, we establish an estimate of the quantity (|zf'/f|) and obtain the region of variability of the function (zf''/f') for (zin {mathbb D}), (fin S(p)). After that, we define radius of concavity and compute the same for functions in S(p) and for some other well-known classes of functions. We also explore linear combinations of functions belonging to S(p) and some other classes of analytic univalent functions and investigate their radii of univalence, convexity and concavity.

The limit q-Durrmeyer operator, (D_{infty ,q}), was introduced and its approximation properties were investigated by Gupta (Appl. Math. Comput. 197(1):172–178, 2008) during a study of q-analogues for the Bernstein–Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of (D_{infty ,q}). The interrelation between the analytic properties of a function f and the rate of growth for (D_{infty ,q}f) are established, and the sharpness of the obtained results are demonstrated.

We study connections between a new type of linear differential inequalities and normality or quasi-normality. We prove that if (C>0), (kge 1) and (a_0(z),dots ,a_{k-1}(z)) are fixed holomorphic functions in a domain D, then the family of the holomorphic functions f in D, satisfying for every (zin D)

is quasi-normal in D. For the reversed sign of the inequality we show the following: Suppose that (A,Bin {{mathbb {C}}}), (C>0) and (mathcal {F}) is a family of meromorphic functions f satisfying for every (zin D)

and also at least one of the families (left{ f'/f:fin mathcal {F}right} ) or (left{ f''/f:fin mathcal {F}right} ) is normal. Then (mathcal {F}) is quasi-normal in D.

We show that for minimal graphs in (R^3) having 0 boundary values over simpy connected domains, the maximum over circles of radius r must be at least of the order (r^{1/2}).

Abstract

We investigate some topological properties of Julia components, that is, connected components of the Julia set, of a transcendental entire function f with a multiply-connected wandering domain. If C is a Julia component with a bounded orbit, then we show that there exists a polynomial P such that C is homeomorphic to a Julia component of the Julia set of P. Furthermore if C is wandering, then C is a buried singleton component. Also we show that under some dynamical conditions, every such C is full and a buried component. The key for our proof is to show that some iterate of f can be regarded as a polynomial-like map on a suitable arbitrarily large bounded topological disk. As an application of this result, we show that a transcendental entire function having a wandering domain with a bounded orbit cannot have multiply-connected wandering domains.