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)

Suppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that \(E'(z) = \pm 1\) whenever \(E(z) = 0\). Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of

We some open questions linked to the notion of neighbourhood of a univalent function as introduced by Stephan Ruscheweyh in 1981.

We consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in \(({{\mathbb {R}}}^+)^d\). We define the logarithmic indicator function on \({{\mathbb {C}}}^d\): $$\begin{aligned} H_P(z):=\sup _{ J\in P} \log |z^{ J}|:=\sup _{ J\in P} \log [|z_1|^{ j_1}\cdots |z_d|^{ j_d}] \end{aligned}$$ and an associated class of plurisubharmonic (psh) functions:

We determine \(\displaystyle \sup \left\{ \frac{|(1-z)^\alpha -(1-w)^\alpha |}{|z-w|^\alpha }: |z|,|w|\le 1,z\not =w\right\} \) where \(0<\alpha \le 1\).

The Lambert W function is the multi-valued inverse of the function \(E(z) = z \exp z\). Let \(\widetilde{W}\) be a branch of W defined and single-valued on a region \(\widetilde{D}\). We show how to use the Taylor expansion of \(\widetilde{W}\) at a given point of \(\widetilde{D}\) to obtain an infinite series representation of \(\widetilde{W}\) throughout \(\widetilde{D}\).

We prove that the family \(\mathcal {F}_C(D)\) of all meromorphic functions f on a domain \(D\subseteq \mathbb {C}\) with the property that the spherical area of the image domain f(D) is uniformly bounded by \(C \pi \) is quasi-normal of order \(\le C\). We also discuss the close relations between this result and the well-known work of Brézis and Merle on blow-up solutions of Liouville’s equation.

In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

In this paper, we describe entire solutions for two certain types of non-linear differential-difference equations of the form $$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)}, \end{aligned}$$ and $$\begin{aligned} f^n(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=p_1e^{\lambda z}+p_2e^{-\lambda z}, \end{aligned}$$ where q, Q, u, v are non-constant polynomials, \(c,\lambda

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and \(f(z+\eta )\) share a, b, c IM, where \(\eta \) is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then \(f(z)=f(z+\eta )\) for all \(z\in \mathbb {C}\). We prove that if a non-constant finite order entire function

For a given ring (domain) in \(\overline{\mathbb {R}}^n\), we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all \(n\ge 3\), the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the boundedness of the moduli of the separating rings. We also

An exponential polynomial of order q is an entire function of the form $$\begin{aligned} f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, \end{aligned}$$ where the coefficients \(P_j(z),Q_j(z)\) are polynomials in z such that $$\begin{aligned} \max \{\deg (Q_j)\}=q. \end{aligned}$$ In 1977 Steinmetz proved that the zeros of f lying outside of finitely many logarithmic strips around so called critical

Let \(1\le p<\infty \), and \(\alpha >0\). Let \(F_{\alpha }^{p}\) denote the Fock space. We establish some sharp pointwise estimates for the derivatives of the functions belonging to \(F_{\alpha }^{p}\). Moreover for the Hilbert case \(p=2\) we establish some more specific pointwise sharp estimates. We also consider the differential operator between \(F_{\alpha }^{p}\) and \(F_{\beta }^{p}\) for \(\beta

Let \(\varTheta _{3} (z):= \sum \nolimits _{n\in \mathbb {Z}} \exp (\mathrm {i} \pi n^2 z)\) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane \(\mathbb {H}:=\{z\in \mathbb {C}\,|\,\,\mathrm{{Im}}\, z>0\}\), and takes positive values along \( \mathrm {i} \mathbb {R}_{>0}\), the positive imaginary axis, where \(\mathbb {R}_{>0}:= (0, +\infty )\). We define

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications

In this article we estimate the sum of coefficients for functions with restrictions on the pre-Schwarzian derivative. We obtain an estimate, which is sharp up to a constant and find upper and lower bounds for that constant. Also we estimate the sum of the first three coefficients for all functions f, such that \(\log f'(z)\) is bounded with respect to the Bloch norm.

It is well-known that the Floater-Hormann rational interpolants give better results than other rational interpolants, especially in convergence rates and barycentric form. In this paper, we propose and study a family of bivariate Floater-Hormann rational interpolants, which have no real poles and arbitrarily high convergence rates on any rectangular region. Moreover, these interpolants are linear on

Several necessary and sufficient conditions for a family of Möbius maps to be a normal family in the extended complex plane \(\mathbb {C}_\infty \) are established. Each of these conditions involves collections of two or three points which may vary with the Möbius maps in the family, provided the points satisfy a uniform separation condition. In addition, we derive a sufficient condition for the normality

In terms of pre-Schwarzian and Schwarzian derivatives, we obtain some sufficient conditions to ensure that a sense-preserving locally univalent harmonic mapping f in the unit disk can be extended to a quasiconformal mapping in the complex plane so that its complex dilatation satisfies some Carleson measure conditions.

In this paper, we discuss the Pólya–Szegő continuous symmetrization and its applications to isoperimetric inequalities. In particular, we survey results concerning monotonicity properties of certain characteristics, including torsional rigidity of cylindrical beams and principal frequency of a uniformly stretched elastic membrane of a drum, of triangles and other domains. Several remaining open problems