Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the

For a domain \( D \subsetneq {\mathbb {R}}^{n} \), Ibragimov’s metric is defined as $$\begin{aligned} u_{D}(x,y) = 2\, \log \frac{|x-y|+\max \{d(x),d(y)\}}{\sqrt{d(x)\,d(y)}}, \quad \quad x,y \in D, \end{aligned}$$ where d(x) denotes the Euclidean distance from x to the boundary of D. In this paper, we compare Ibragimov’s metric with the classical hyperbolic metric in the unit ball or in the upper

The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity

Let \(\Omega \) be a smooth, bounded, convex domain in \({\mathbb {R}}^n\) and let \(\Lambda _k\) be a finite subset of \(\Omega \). We find necessary geometric conditions for \(\Lambda _k\) to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling

We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem for a quasi proximate order, i.e. a counterpart of Valiron’s theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain \(D\subset {\mathbb {C}}^m\) into \({\mathbb {P}}^n\) to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto’s work. The approach

In this paper, we deal with the generalized Grötzsch ring function \(\mu _a(r)\) for \(r\in (0,1)\) in the theory of the Ramanujan generalized modular equation and present new inequalities for \(\mu _a(r)\).

In this paper we prove a Radó type result showing that there is no univalent polyharmonic mapping of the unit disk onto the whole complex plane. We also establish a connection between the boundary functions of harmonic and biharmonic mappings. Finally, we show how a close-to-convex biharmonic mapping can be constructed from a convex harmonic mapping.

The positive solutions of the equation \(x^y = y^x\) have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation \(x^y=y^x\), the complex equation \(z^w = w^z\) has virtually been ignored in the literature. In this

The unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type \(\tau >0\) at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf

The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the

In this note we present an alternative to Ford’s construction of the isometric circle of a Möbius map. This construction is based on the double coset decomposition of a group, together with the action of Möbius maps on spherical and hyperbolic spaces.

For a transcendental entire function f, the property that there exists \(r>0\) such that \(m^n(r)\rightarrow \infty \) as \(n\rightarrow \infty \), where \(m(r)=\min \{|f(z)|:|z|=r\}\), is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus

In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with

If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path \(\gamma \) from 0 to infinity such that \(f(z) - a\) tends to 0 as z tends to infinity along \(\gamma \). The Denjoy–Carleman–Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2, mean type. A long-standing problem asks

In an earlier paper, Dorff et al., considered the classes of harmonic univalent functions \(f_k=h_k+\overline{g_k}, k=1,2\), that are shears of \(h_k-g_k=\frac{1}{2} \log [\left( 1+z\right) / \left( 1-z\right) ]\) with dilatations \(\omega _k=e^{i\theta _k}z^{n_k}\) for \(n_k\) positive integers, and proved that if the convolution \(f_1*f_2= h_1*h_2+\overline{g_1*g_2}\) is locally one-to-one and sense-preserving

We develop a numerical solver for three-dimensional poroelastic wave propagation, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem; we also consider attenuation mechanisms both in Biot’s low-

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let \(E\not \subset {\hat{\mathbb {R}}}\) be an arc or a circle. If a map \(f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}\) preserves cross-ratios in E, then f is a Möbius transformation

In this paper, we consider transcendental meromorphic solutions f of finite order \(\rho \) and few poles in the sense that \(S_{\lambda }(r,f):=O(r^{\lambda +\varepsilon })\), where \(\lambda <\rho \) and \(\varepsilon \in (0,\rho -\lambda )\), of the delay-differential equation $$\begin{aligned} f^n+L(z,f)=p_1(z)e^{\alpha _{1}(z)}+p_2(z)e^{\alpha _{2}(z)}, \end{aligned}$$ where \(n\ge 2\) is an integer

We establish generic existence of Universal Taylor Series on products \(\varOmega = \prod \varOmega _i\) of planar simply connected domains \(\varOmega _i\) where the universal approximation holds on products K of planar compact sets with connected complements provided \(K \cap \varOmega = \emptyset \). These classes are with respect to one or several centers of expansion and the universal approximation

This paper deals with the so-called Radon inversion problem formulated in the following way: Given a \(p>0\) and a strictly positive function H continuous on the unit circle \({\partial {\mathbb {D}}}\), find a function f holomorphic in the unit disc \({\mathbb {D}}\) such that \(\int _0^1|f(zt)|^pdt=H(z)\) for \(z \in {\partial {\mathbb {D}}}\). We prove solvability of the problem under consideration

We propose a system of non-linear equations equivalent to the fifth Painlevé equation, which enables us to examine the general singular solution given by Andreev and Kitaev along the positive real axis. We present a two-parameter family of asymptotic solutions corresponding to this general singular solution, and pose a conjecture.

We say that two meromorphic functions f and g share a small function \(\alpha \) counting multiplicities if \(f-\alpha \) and \(g-\alpha \) admit the same zeros with the same multiplicities. Let Q be a polynomial of one variable. In [Comput. Methods Funct. Theory 17: 613-634, 2017, Theorem. 1.1], we proved that if \((Q(f))^{(k)}\) and \((Q(g))^{(k)}\) share \(\alpha \) counting multiplicities then

We improve a normality result of Liu–Li–Pang [4] concerning shared values between two families. Let \(\mathcal F\) and \(\mathcal G\) be two families of meromorphic functions on D whose zeros are multiple. Suppose that \(\mathcal G\) is normal on D, and no sequence contained in \(\mathcal G\) \(\chi \)-converges locally uniformly to \(\infty \) or a function g satisfying \(g'\equiv 1\). If for every

We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic system, which is analyzed using the method of real comprehensive triangular decomposition.

A continuous complex-valued function F in a domain \(D\subseteq \mathbf {C}\) is poly-analytic of order \(\alpha \) if it satisfies \(\partial ^{\alpha }F/\partial \overline{z}^{\alpha }=0\). One can show that F has the form \(F(z)=\sum _{k=0}^{\alpha -1}\overline{z}^{k}A_{k}(z)\), where each \(A_k\) is an analytic function. In this paper, we prove the existence of a Landau constant for poly-analytic

The authors present sharp transformation inequalities for the zero-balanced hypergeometric function \({}_2F_1(a,b;a+b;r)\) created by the transformations \(r\mapsto x=[(1-r)/(1+2r)]^3\) and \(r\mapsto 1-x\), \(r\mapsto u=(1/2)r(3+r)^2(1+r)^{-3}\) and \(r\mapsto 1-u\), \(r\mapsto p=(27/2)r(1+r)^4(1+4r+r^2)^{-3}\) and \(r\mapsto 1-p\), by showing the monotonicity properties of certain combinations in

In this paper, we consider functions analytic in the unit disk that are subordinate to functions of the same type that are defined by certain differential subordinations. We prove several sharp majorization theorems and a product theorem.

In this paper, making use of the value distribution theory for meromorphic functions in several complex variables and its difference analogues, we mainly consider the uniqueness problem for meromorphic functions in several complex variables sharing values or small functions with their shifts or difference operators, and weaken the condition of growth of hyper-order of meromorphic functions, which generalize

The aim of this paper is to establish some properties of solutions to the Dirichlet–Neumann problem: \((\partial _z\partial _{\overline{z}})^2 w=g\) in the unit disc \({{\mathbb {D}}}\), \(w=\gamma _0\) and \(\partial _{\nu }\partial _z\partial _{\overline{z}}w=\gamma \) on \({\mathbb {T}}\) (the unit circle), \(\frac{1}{2\pi i}\int _{{\mathbb {T}}}w_{\zeta {\overline{\zeta }}}(\zeta )\frac{d\zeta

In this paper, some characterizations for the compact difference of composition operators on weighted Bergman spaces \(A^p_\omega \) with doubling weights are given, which extend Moorhouse’s characterization for the difference of composition operators on the weighted Bergman space \(A^2_\alpha \).

A correction to this paper has been published: https://doi.org/10.1007/s40315-021-00387-4

Let X be a compact subset of the complex plane with the property that every relatively open subset of X has positive area and let \(A_0(X)\) denote the space of VMO functions that are analytic on X. \(A_0(X)\) is said to admit a bounded point derivation of order t at a point \(x_0 \in \partial X\) if there exists a constant C such that \(|f^{(t)}(x_0)|\le C \Vert f\Vert _{{\text {BMO}}}\) for all functions

For a quasiregular mapping \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\), with \(n\ge 2\), a Julia limiting direction \(\theta \in S^{n-1}\) arises from a sequence \((x_n)_{n=1}^{\infty }\) contained in the Julia set of f, with \(|x_n| \rightarrow \infty \) and \(x_n/|x_n| \rightarrow \theta \). Julia limiting directions have been extensively studied for entire and meromorphic functions in the

The paper deals with the problem of convergence of special classes of branched continued fractions including the multidimensional A- and J-fractions with independent variables, which are generalizations of A-fractions (associated fractions) and J-fractions (Jakobi’s fractions), respectively. These branched continued fractions are used for the approximation of analytic functions of several complex variables

In the present paper, we study the order of convexity of \(z{_2F_1}(a,b;c;z)\) with real parameters a, b and c where \({_2F_1}(a,b;c;z)\) is the Gaussian hypergeometric function. First we obtain some conditions for \(z{_2F_1}(a,b;c;z)\) with no finite orders of convexity by considering its asymptotic behavior around \(z=1\). Then the order of convexity of \(z{_2F_1}(a,b;c;z)\) is demonstrated for some

We find explicit forms for all the entire solutions of a certain type of non-linear binomial differential equation. This has connections to results of Hayman, Mues, Langley and Bergweiler. Observations about entire solutions of two similar types of binomial differential equations are discussed, and open questions about all three types of equations are posed.

For pairs of holomorphic maps \((u,\psi )\) on the complex plane, we study some dynamical properties of the weighted composition operator \(W_{(u,\psi )}\) on the Fock spaces. We prove that no weighted composition operator on the Fock spaces is supercyclic. Conditions under which the operators satisfy the Ritt’s resolvent growth condition are also identified. In particular, we show that a non-trivial

We study the topology and geometry of the space \(\varUpsilon \) of shapes of abstract tetrahedra, i.e. flat metrics on the 2-sphere with four conical singularities, up to equivalence by orientation-preserving similarities. There is no labeling of the conical singularities; two of them with the same cone-angle can be interchanged by a similarity. We are dealing with the geometric structure introduced

Starting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space dimensions and one time dimension), we complexify the independent variables, thus doubling the number of real variables, and hence we work in 4+2 dimensions: \(x_1\), \(x_2\), \(y_1\), \(y_2\) and \(t_1\), \(t_2\). In this paper we solve the initial value problem of the 3-wave interaction equations in 4+2 dimensions.

New asymptotic relations between the \(L_p\)-errors of polynomials approximation of univariate functions by algebraic polynomials and entire functions of exponential type are obtained for \(p\in (0,{\infty }]\). General asymptotic relations are applied to functions \(\vert x\vert ^{{\alpha }+i{\beta }},\,\vert x\vert ^{{\alpha }}\cos ({\beta }\log \vert x\vert )\), and \(\vert x\vert ^{{\alpha }}\sin

Recently, some uniqueness theorems about meromorphic functions f(z) concerning their derivatives \(f'(z)\) and shifts \(f(z+c)\) with three CM sharing values have been obtained. In this paper, we continue to study this topic. We consider not only high order derivatives instead of just \(f'(z)\), but also IM sharing value instead of CM sharing value. In fact, we mainly prove that for a non-constant

Let \(\Omega \) be a convex domain in the complex plane \({\mathbb C}\) with \(\Omega \not = {\mathbb C}\), and P be a conformal map of the unit disk \({\mathbb D}\) onto \(\Omega \). Let \({\mathcal F}_\Omega \) be the class of analytic functions g in \({\mathbb D}\) with \(g({\mathbb D}) \subset \Omega \). Also, let \(H_1^\infty ({\mathbb D})\) be the well known closed unit ball of the Banach space

Honoring Walter Hayman’s groundbreaking work on entire functions, the cover of this volume is based on his counterexample to Wiman’s conjecture. In this note we sketch some background.

Montel’s fundamental normality test (published in 1912) provides a strong sufficient condition for normality: a family \(\mathscr {F}\) of functions meromorphic in a region \(\varOmega \) is normal there if there exist three distinct values a, b, c in the extended complex plane \({\mathbb {C}}_\infty \) such that each f in \(\mathscr {F}\) omits in \(\varOmega \) each of these values. In 1954 Montel

We study the maximum modulus set, \({{\mathcal {M}}}(p)\), of a polynomial p. We are interested in constructing p so that \({{\mathcal {M}}}(p)\) has certain exceptional features. Jassim and London gave a cubic polynomial p such that \({{\mathcal {M}}}(p)\) has one discontinuity, and Tyler found a quintic polynomial \({\tilde{p}}\) such that \({{\mathcal {M}}}({\tilde{p}})\) has one singleton component

Given two harmonic functions \(f_{1}=h_{1}+{\bar{g}}_{1}\) and \(f_{2}=h_{2}+{\bar{g}}_{2}\) defined on the open unit disk of the complex plane, the geometric properties of the product \(f_{1}\otimes f_{2}\) defined by $$\begin{aligned} f_{1}\otimes f_{2}=h_{1}*h_{2}-\overline{g_{1}*g_{2}}+ i{\text {Im}}(h_{1}*g_{2}+h_{2}*g_{1}) \end{aligned}$$ are discussed. Here \(*\) denotes the analytic convolution

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in \(\mathbb C^k\), \(k \ge 3.\) First, we study the relation between two polynomial shift-like maps in \(\mathbb C^k\), \(k \ge 3\), that share the same forward and backward Julia sets. Secondly, we study the relation between any pair of skew products of Hénon maps in \(\mathbb C^3\) having the

The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, \(f(z)^nf^{(k)}(z)\) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case \(k=1\). When \(k\ge 2\) the conjecture is shown to be true under certain conditions even if n

In this paper, we discuss the conditions for the function $$\begin{aligned} F_a(z) =\sum _{k=0}^\infty \frac{z^k}{(a+1)(a^2+1) \cdots (a^k+1)},\quad a >1, \end{aligned}$$ to belong to the Laguerre–Pólya class, or to have only real zeros.

The aim of this paper is to classify the cubic polynomials $$\begin{aligned} H(z,x,y)=\sum _{j+k\le 3}a_{jk}(z)x^jy^k \end{aligned}$$ over the field of algebraic functions such that the corresponding Hamiltonian system \(x'=H_y,\) \(y'=-H_x\) has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several non-trivial Hamiltonians which are

In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If \(f'(z)\) and \(f(z+c)\) share a, b IM, then \(f'(z)\equiv f(z+c)\). This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178

In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative \(f^{(k)}\), \(k \ge 1\). Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999).

In this paper, by using the formula of Faà di Bruno for the k-th derivative of a composite function, we establish a refinement of the Fekete and Szegő inequality for a class of holomorphic mappings on bounded starlike circular domains in \(\mathbb {C}^n\). The results presented here generalize some known results. Finally, a certain problem is also proposed.

In the article, we present the best possible bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete p-elliptic integrals.

In this work, we give some uniqueness theorems for non-constant zero-order meromorphic functions when they and their q-shifts partially share values in the extended complex plane. This is a continuation of previous works of Charak et al. (J Math Anal Appl 435(2):1241–1248, 2016) and of Lin et al. (Bull Korean Math Soc 55(2):469–478, 2018). Furthermore, we show some uniqueness results in the case multiplicities

This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy

Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$(1) where \(W(A):=\{\langle Ax,x\rangle : \Vert x\Vert =1\}\) denotes the numerical range of A. This would then also hold for all functions f which are analytic in a neighborhood of W(A)