Let G=ℝ^2 - { 0 } be equipped with the point pair function as a metric. We obtain several properties of equidistant circles and prove that the equilateral dimension of G is 4.
We study Schauder bases for spaces of holomorphic functions in the open unit disk 𝔻 . Given a non-Blaschke sequence (λ _n)_n≥ 1 in 𝔻 , we show that the associated sequence of finite Blaschke products (B_n)_n≥ 1 forms a Schauder basis for Hol(𝔻) when this space is endowed with a norm inherited from a Banach space X satisfying a set of natural structural assumptions. This abstract framework includes, in particular, the classical Hardy spaces H^p , 1≤ p≤∞ , the weighted Bergman spaces A_α ^p , 1≤ p≤∞ , α >-1 , and BMOA. We further prove that if the sequence (λ _n)_n≥ 1 is contained in a compact subset of the open unit disk, then (B_n)_n≥ 1 is a Schauder basis for Hol(𝔻) endowed with its natural topology. Moreover, in the case of Hardy spaces H^p , 1
Let G be a finite Jordan domain in the complex plane ℂ , bounded by a Carleson curve Γ and let ω be a weight function given on Γ . In this work some direct and inverse theorems of approximation theory in the weighted variable exponent Smirnov classes E_ω^p(· )(G) , under some restrictions on the weight ω and variable exponent p(· ) , are proved.
Honoring the 200th birthday of Bernhard Riemann on September 17, the cover of this volume shows a phase plot of the Riemann Zeta Function. In this essay we present some features of this function in an entertaining way. Using only basic tools, we try to make it clear that Zeta is, on the one hand, very "simple" and, on the other hand, incredibly complex. In particular, we give an interpretation of its universality and illustrate this property by phase plots. The last section is devoted to philosophical speculations on the Riemann Hypothesis.
In this paper, we investigate removability for generalized John metric spaces. We prove that X is a generalized John metric space if and only if X\ P is a generalized John metric space, where P is a countable subset of X which satisfies a quasihyperbolic b-separation condition with parameter b>0 .
We show that every Blaschke sequence containing at most one zero can be transformed into an interpolating Blaschke sequence by using an arbitrarily small perturbation. More precisely, for a Blaschke sequence {z_n}_n=1^∞ containing at most one zero and any ε > 0, there is an interpolating Blaschke sequence {ω _n}_n=1^∞ such that for 1 ≤ n < ∞ , (1) |z_n| = |ω _n|, (2) and | z_n - ω _n| < ε . This result is a refinement of Naftalevič’s Theorem.
Honoring the 200th birthday of Bernhard Riemann on September 17, the cover of this volume shows a phase plot of the Riemann Zeta Function. In this essay we present some features of this function in an entertaining way. Using only basic tools, we try to make it clear that Zeta is, on the one hand, very “simple” and, on the other hand, incredibly complex. In particular, we give an interpretation of its universality and illustrate this property by phase plots. The last section is devoted to philosophical speculations on the Riemann Hypothesis.
In 1990 s, by using the quasihyperbolic metric and quasi-isometric mappings as main tools, Väisälä established the theory of (dimension) freely quasiconformal mappings in real Banach spaces. In this paper, we employ relative distance and relative mappings to characterize the related mapping classes in Väisälä’s free quasiworld, including semisolid mappings, Lipschitz mappings in the quasihyperbolic metric, coarsely Lipschitz mappings in the quasihyperbolic metric, and two-sided conditions about these mapping classes. Next, we study the fully punctured properties of these mapping classes and obtain new characterizations of freely quasiconformal mappings in Banach spaces.
In this work, we study time evolution properties of univalent functions in the context of Hele–Shaw flows. We introduce the notion of circular tolerance and its generalization, the circular tolerance of order β , as new measures of distortion. We also define the concepts of restricted starlikeness and restricted strongly starlikeness. We investigate the time invariance of the properties of restricted starlikeness and restricted strongly starlikeness, both for the interior version of the Hele–Shaw problem. Our results show that, in the case of starlike dynamics and strongly starlike dynamics of order β , the upper bounds for circular tolerance and, respectively, for circular tolerance of order β are preserved, provided that these bounds are at least 1, for the inner version of the Hele–Shaw problem with injection and without surface tension. Several consequences of these results improve and refine previously known results on invariant geometric properties in Hele–Shaw dynamics.
Matrix valued (asymmetric) truncated Toeplitz operators are generally not complex symmetric. In this paper, we define a new conjugation with unique properties and study its relation to matrix valued asymmetric truncated Toeplitz operators. We also explore the connections between matrix valued asymmetric truncated Toeplitz operators and Hankel operators.
In geometric function theory, the problem of finding coefficient bounds plays an important role. In the present paper, we consider the third Hankel determinant defined for the coefficients of a function f belonging to the classes ℳ and 𝒩 of analytic functions associated with the number 3/2. The main aim is to give the sharp upper bounds of the third Hankel determinant for the two classes. The results improved many known outcomes and are the best possible.
Firstly, fundamental properties of the operator D_ω are established, and the decomposition theorem of the polynomial weighted Dirac operator D_ω^k is given. Secondly, decomposition theorems of polynomial weighted Dirac operators are obtained. Finally, as the applications of the above decomposition theorems, two kinds of generalized Riquier boundary value problems are solved.
Zero inclusion sectors are obtained for a family of complex valued harmonic rational functions. The zeros of a limiting case are obtained explicitly.
In this paper we consider local inverse estimates for singular integrals with the Hilbert kernel. This gives the unimprovability of a result from Mamedkhanov and Jafarov (Ukr Math J 75(5):703–718, 2023).
In this paper, we investigate several properties of a harmonic mapping f=𝒫[F] on the unit disk with boundary function F∈L^p(T) and p∈[1, ∞ ] . The coefficients of f are initially estimated, and these estimates are directly employed in analysis of the Landau theorem. Subsequently, the Schwarz lemma for the harmonic mapping f is established by applying certain properties of Gauss hypergeometric functions. This work generalizes the classical Schwarz lemma to bounded harmonic mappings. As an application, two new versions of the Schwarz-Pick lemmas for the harmonic mapping f are also discussed.
We construct group-invariant CR maps from the unit sphere in ℂ^3 and provide bounds for the gap termination in this setting.
Recent researchers have investigated how the zeros of certain families of complex harmonic functions change with a single parameter. Many leverage the well-behaved images of the critical curve and the harmonic analogue of the Argument Principle to prove zero-counting theorems. In this paper, we investigate the zeros of a family of harmonic functions for which the image of its critical curve is a non-singular linear image of an epicycloid. By analyzing this curve and using the harmonic analogue of the Argument Principle, we obtain a detailed zero-counting theorem for our family.
This paper is a sequel to our work in [11]. Here, we primarily study the dynamics of the adjoint of a weighted forward shift operator F_w on the analytic function space ℓ ^p_a,b having a normalized Schauder basis of the form {(a_n+b_nz)z^n:n ≥ 0} . We obtain sufficient conditions for F_w to be continuous, and show, under certain conditions, that the operator F_w is similar to a compact perturbation of a weighted forward shift on ℓ ^p(ℕ_0) . This also allows us to obtain the essential spectrum of F_w . Further, we study when the adjoint F_w^* is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on ℓ ^p(ℕ_0) . Finally, it is proved that the adjoint of a shift on the dual of ℓ ^p_a,b can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero–one law of orbital limit points fails for F_w^* , which means that, under certain conditions, the adjoint F_w^* is non-hypercyclic, but it has an orbit possessing non-zero norm limit points.
The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Prop. 4.5, Prop. 4.13, Astérisque 270 (2001)].
This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are based on pairs of Fredholm operator valued functions. Some factorization and isometric embedding results are extended from the scalar valued theory of de Branges spaces. In particular, global factorization of Fredholm operator valued entire functions and analytic equivalence of reproducing kernels of de Branges spaces are discussed. Additionally, the operator valued entire functions associated with these de Branges spaces are studied, and a connection with the operator nodes is established.
