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

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

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

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