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

We prove that all solutions \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}\) of the functional inequality $$\begin{aligned} (*) \quad f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y), \end{aligned}$$ which are convex or concave on \({\mathbb {R}}\) and differentiable at 0 are given by $$\begin{aligned} f(x)=x\quad \text{ and } \quad f(x)\equiv c, \quad \text{ where } \quad 0\le c\le 2. \end{aligned}$$ Moreover, we

Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr’s

Let \(\sigma \) be a Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of the orthogonal matrix polynomials associated with the Christoffel matrix transformation of \(\sigma \) defined by $$\begin{aligned} d\sigma _{c_m}(z)=W_m(z)^Hd\sigma (z)W_m(z), \end{aligned}$$ where \(W_m(z)=\prod _{j=1}^m(z\mathbf{I} -A_j)\) and \(A_j\)

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space \({\mathbb {R}}^{2n}\) of even dimension

This article introduces the F-classes of the convex, starlike and close-to-convex classes of univalent functions. The paper also extends the theory of differential subordinations to F-differential subordinations and presents examples of these new concepts.

Differential equations of the form \(f'' + A(z)f' + B(z)f = 0\) (*) are considered, where A(z) and \(B(z) \not \equiv 0\) are entire functions. The Lindelöf function is used to show that for any \(\rho \in (1/2, \infty )\), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying \(\rho =\rho (f)\ge \rho (A)\ge \rho (B)\), where \(\rho

The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on \({\mathbb {R}}^3\) (sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic

For \(0

We study certain integral type operators acting on a large class of Banach spaces of analytic functions on the open unit disc. The operators map into weighted Banach spaces of analytic functions, Bloch type spaces or Zygmund type spaces. Besides characterizing boundedness, we give essential norm estimates of these operators. In order to investigate these operators, we also study weighted differentiation

Let \(p_1, p_2\) and \(\alpha _1, \alpha _2\) be non-zero constants, and \(P_d(z, f)\) be a differential polynomial in f of degree d. Li obtained the forms of meromorphic solutions with few poles of the non-linear differential equations \(f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}\) provided \(\alpha _1\ne \alpha _2\) and \(d\le n-2\). In this paper, given \(d=n-1\), we find the forms of meromorphic

In this paper, we study the boundedness and compactness of the identity operator \(I:BMOA_{\log }\rightarrow \mathcal {T}^{\infty }_{\log }(\mu )\). As applications, we characterize the boundedness and compactness of the Volterra integral operators \(T_g\) and \(I_g\) on the space \(BMOA_{\log }\). The estimations for the essential norm of \(T_g\) and \(I_g\) on the space \(BMOA_{\log }\) are also

In this paper, we discover new connections between topics in classical geometric function theory and complex dynamics. In particular, we study some classes of starlike functions and their embeddings in the classes of semigroup generators. In addition, we find explicit formulas for the radii of starlikeness for those classes and uniform rates of convergence of semigroups to their Denjoy–Wolff points

Let \(D\in \mathbb {N}\), \(q\in [2,\infty )\) and \((\mathbb {R}^D,|\cdot |,dx)\) be the Euclidean space equipped with the D-dimensional Lebesgue measure. In this article, we establish the Fefferman–Stein decomposition of Triebel–Lizorkin spaces \(\dot{F}^0_{\infty ,\,q'}(\mathbb {R}^D)\) with the help of the dual on function sets which have special topological structure. A function in Triebel–Lizorkin

We construct a boundary integral formula for harmonic functions on smoothly-bordered subdomains of Riemann surfaces embeddable into \({\mathbb {C}}{\mathbb {P}}^2\). The formula may be considered as an analogue of the Green’s formula for domains in \({\mathbb {C}}\).

In the original publication, article title was incorrectly published as.

In this paper, by using Belyi maps and dessin d’enfants, we construct some concrete examples of Strebel differentials with four double poles of residues 1, 1, 1, 1 on the Riemann sphere. We also prove that they have either two double zeroes or four simple zeroes. In particular, we show that they have two double zeroes if and only if their poles are coaxial; in such cases we also obtain their explicit

Let \(\{\varphi _k\}_{k=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study the variance of the number of zeros of random linear combinations of the form$$\begin{aligned} P_n(z)=\sum _{k=0}^{n}\eta _k\varphi _k(z), \end{aligned}$$where \(\{\eta _k\}_{k=0}^n \) are complex-valued random variables. Under the assumption that

The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumann–Carathéodory rigidity constant, the higher order curvatures of the Carathéodory metric and a conformal metric arising from holomorphic quadratic differentials.

This work is continuation of a recent paper (Qi et al. in Comput Methods Funct Theory 18:567–582, 2018). Here, we obtain three uniqueness results of the derivative of f(z) with its shift \(f(z+c)\), leading to a new way of studying the complex differential-difference equation \(f'(z)=f(z+c)\).

Let \(a(\ne 0)\), b be finite complex numbers, and \(k\ge 3\) be a positive integer. If f is a meromorphic function of order \(\rho >2\), then \(f^{\prime }-af^k-b\) has infinitely many zeros on the complex plane.

We propose a sufficient condition for the convergence of a complex power type formal series of the form \(\varphi =\sum _{k=1}^{\infty }\alpha _k(x^{\mathrm{i}\gamma })\,x^k\), where \(\alpha _k\) are functions meromorphic at the origin and \(\gamma \in {{\mathbb {R}}}\setminus \{0\}\), that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of such a type formal

We establish two sharp inequalities involving the power mean and generalized elliptic integral of the first kind. As applications, the analogous inequalities concerning the complete p-elliptic integral of the first kind are also derived.

Let \(f(z) := \sum f_\nu z^\nu \) be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Padé approximants \(\{\pi _{n,m_n}\}\) associated with f, where \(m_n\rightarrow \infty \), \(m_n\le m_{ n+1}\le m_n+1\) and \(m_n = o(n/\log n)\), resp. \(m_n = o(n)\) as \(n\rightarrow \infty \). We extend classical results

In this paper, we will give characterizations for weighted Bergman spaces with admissible weights. The first criterion is established by the invariant gradient on the unit ball of \({\mathbb {C}}^N\). The other criterion is given by the radial derivative of holomorphic functions in the unit ball.

In the present paper we study the boundary behavior of a weighted Koppelman type integral with a specific choice of weight for a function \(\phi \) that is integrable on a bounded domain \(D\subset \mathbb {C}^n\) and is continuous on its \(\mathcal {C}^1\)-boundary. Applying the above results, we derive a variation of Hartogs phenomena about the holomorphicity of a function \(\phi \) which is integrable

Honoring Stephan Ruscheweyh’s magnificent contributions to complex analysis in general and to geometric function theory in particular, the cover of this volume shows a (modified) phase plot of a Ruscheweyh derivative of the complex tangent function. In this note we sketch some background.

Let \({ G\subset {\mathbb {C}} }\) be a Jordan domain with rectifiable Dini smooth boundary \(\varGamma \). In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In

In this article we prove Bohr inequalities for sense-preserving K-quasiconformal harmonic mappings defined in the unit disk \({{\mathbb {D}}}\) and obtain the corresponding results for sense-preserving harmonic mappings. In addition, Bohr inequalities are established for uniformly locally univalent holomorphic functions, and for \(\log (f(z)/z)\) where f is univalent or inverse of a univalent function

Let f be analytic in the unit disk \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}\), and \({\mathcal {S}}\) be the subclass of normalized univalent functions given by \(f(z)=z+\sum _{n=2}^{\infty }a_n z^n\) for \(z\in {\mathbb {D}}\). We give bounds for \(| |a_3|-|a_2| | \) for the subclass \({\mathcal B}(\alpha ,i \beta )\) of generalized Bazilevič functions when \(\alpha \ge 0\), and \(\beta \) is

Exponential type functions are important subclasses of transcendental entire functions. In this paper, we will use some results given by Steinmetz (Manuscr Math 26:155–167, 1978) to consider the zeros of difference or differential-difference polynomials of exponential polynomials. In addition, we also consider the zeros of difference polynomials of exponential type functions with infinite order.

In this article, we give an equivalent characterization for close to almost starlike mappings of order \(\alpha \)\((0\le \alpha <1)\) in terms of Loewner chains. Next, the growth theorem and distortion theorem along a direction are obtained for this subclass of biholomorphic mappings. In particular, the results of Pfaltzgraff and Suffridge can be obtained when \(\alpha =0\).

In this article, by giving a new method to estimate the counting functions of the auxiliary function, we prove a new uniqueness theorem for degenerate meromorphic mappings sharing moving hyperplanes regardless of multiplicity. Our result extends and improves almost all results in this topic.

The Baran metric \(\delta _E\) is a Finsler metric on the interior of \(E\subset {\mathbb {R}}^n\) arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, \(\delta _E\) is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltrami operator associated with \(\delta _E\) are the orthogonal

Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal \(L_\infty \)- and \(L_1\)-norm on K, respectively. This article presents numerical procedures based on semidefinite programming to compute these polynomials in case K is a finite union of compact intervals. For Chebyshev polynomials of the first kind, the procedure makes use of a characterization of polynomial

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

In this article, we investigate the value distribution of L-functions in the (extended) Selberg class and establish two theorems which show how an L-function and a meromorphic function are uniquely determined by their sharing two finite sets. Our results answer a question of Lin and Lin (Filomat 30:3795–3806, 2016). Examples are also given in support of the accuracy of the results.

Let M be a complete and connected Kähler manifold whose universal covering is biholomorphic to a ball in \({\mathbb {C}}^m\). In this article, we investigate algebraic dependence of three meromorphic mappings from M into \({\mathbf {P}}^n({\mathbb {C}})\) sharing hyperplanes in subgeneral position. In addition, we study linear degenerates of the map \(f^1\times f^2 \times f^3\) where \(f_1, f_2\) and

In this paper, we provide a concrete criterion for sets lying in a hyperplane to be removable for weighted Orlicz–Sobolev spaces. We define porous sets and show that the porous sets lying in a hyperplane are removable; this is a generalization of the results in Karak (Potential Anal 43(4):675–694, 2015), Futamura and Mizuta (Hiroshima Math J 33:43–57, 2003).