The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,\,b} \colon z \mapsto ae^z+be^{-z}$ for $a,\,b\in \mathbb{C}^\ast$ , consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,\,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$ . We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$ , and conclude that no two of its dynamic rays land together.

We consider an analogue of Kontsevich’s matrix Airy function where the cubic potential $\textrm{Tr}(\Phi^3)$ is replaced by a quartic term $\textrm{Tr}\!\left(\Phi^4\right)$ . Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.

Singular directions in a Veech surface are shown to be exactly the directions of its saddle connections.

We prove that for any infinite sets of nonnegative integers $\mathcal {A}$ and $\mathcal {B}$ , there exist transcendental analytic functions $f\in \mathbb {Z}\{z\}$ whose coefficients vanish for any indexes $n\not \in \mathcal {A}+\mathcal {B}$ and for which $f(z)$ is algebraic whenever z is algebraic and $|z|<1$ . As a consequence, we provide an affirmative answer for an asymptotic version of Mahler’s problem A.

Inspired by Xiao’s work on Hankel measures for Hardy and Bergman spaces [‘Pseudo-Carleson measures for weighted Bergman spaces’. Michigan Math. J. 47 (2000), 447–452], we introduce Hankel measures for Fock space $F^p_\alpha $ . Given $p\ge 1$ , we obtain several equivalent descriptions for such measures on $F^p_\alpha $ .

We characterize zero sets for which every subset remains a zero set too in the Fock space $\mathcal {F}^p$ , $1\leq p<\infty $ . We are also interested in the study of a stability problem for some examples of uniqueness set with zero excess in Fock spaces.

Two boundary value problems are solved for potential steady-state 2D Darcian seepage flows towards a line sink in a homogeneous isotropic soil from a ponded land surface, which is not flat but profiled. The aim of this shaping is ‘uniformisation’ of the velocity and travel time between this surface and a horizontal drain modelled by a line sink. The complex potential domain is a half-strip, which is mapped onto a reference plane. Either the velocity magnitude or a vertical coordinate along the land surface are control variables. Either a complexified velocity or complex physical coordinate is reconstructed by solving mixed boundary-value problems with the help of the Keldysh-Sedov formula via singular integrals, the kernel of which are the control functions. The flow nets, isotachs and breakthrough curves are found by computer algebra routines. A designed soil hump above the drain ameliorates an unwanted ‘preferential flow’ (shortcut) and improves leaching of salinised soil of a cropfield during a pre-cultivation season.

Let $f(z)=\sum _{n=0}^\infty a_n z^n$ be an entire function on the complex plane, and let ${\mathcal R} f(z) = \sum _{n=0}^\infty a_n X_n z^n$ be its randomization induced by a standard sequence $(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions $f(z)$ such that ${\mathcal R} f(z)$ is almost surely in the Fock space ${\mathcal F}_{\alpha }^p$ for any $p, \alpha \in (0,\infty )$. Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space ${\mathcal F}(\infty , q, \alpha )$, an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.

It is known that, in the unit disc as well as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc ,A_{k-2}$ of

$$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$

$D(0,R)$

$0<R\leqslant \infty $

In this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.

The aim of this paper is to establish the boundedness of fractional type Marcinkiewicz integral $\mathcal {M}_{\iota ,\rho ,m}$ and its commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ on generalized Morrey spaces and on Morrey spaces over nonhomogeneous metric measure spaces which satisfy the upper doubling and geometrically doubling conditions. Under the assumption that the dominating function $\lambda $ satisfies $\epsilon $ -weak reverse doubling condition, the author proves that $\mathcal {M}_{\iota ,\rho ,m}$ is bounded on generalized Morrey space $L^{p,\phi }(\mu )$ and on Morrey space $M^{p}_{q}(\mu )$ . Furthermore, the boundedness of the commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ generated by $\mathcal {M}_{\iota ,\rho ,m}$ and regularized $\mathrm {BMO}$ space with discrete coefficient (= $\widetilde {\mathrm {RBMO}}(\mu )$ ) on space $L^{p,\phi }(\mu )$ and on space $M^{p}_{q}(\mu )$ is also obtained.

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [ 35 ]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

In this paper, we prove the well-known Erdős–Lax inequality [4] in a sharpened form. As a consequence, another widely used inequality due to Ankeny and Rivlin [1] gets sharpened. These results may be useful in various applications that required the Erdős–Lax and the Ankeny–Rivlin inequalities.

In this paper, we give a complete description of closed ideals of the Banach algebra $\mathcal {B}^{s}_{p}\cap \lambda _{\alpha }$ , where $\mathcal {B}^{s}_{p}$ denotes the analytic Besov space and $\lambda _{\alpha }$ is the separable analytic Lipschitz space. Our result extends several previous results in Bahajji-El Idrissi and El-Fallah (2020, Studia Mathematica 255, 209–217), Bouya (2009, Canadian Journal of Mathematics 61, 282–298), and Shirokov (1982, Izv. Ross. Akad. Nauk Ser. Mat. 46, 1316–1332).

Since 1984, many authors have studied the dynamics of maps of the form $\mathcal{E}_a(z) = e^z - a$ , with $a > 1$ . It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions.

It is rather surprising that many of the interesting dynamical properties of the maps $\mathcal{E}_a$ actually arise from their elementary function theoretic structure, rather than as a result of analyticity. We show this by studying a large class of continuous $\mathbb{R}^2$ maps, which, in general, are not even quasiregular, but are somehow analogous to $\mathcal{E}_a$ . We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of $\mathcal{E}_a$ , including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.

Given a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

Let u and $\varphi $ be two analytic functions on the unit disk D such that $\varphi (D) \subset D$ . A weighted composition operator $uC_{\varphi }$ induced by u and $\varphi $ is defined on $A^2_{\alpha }$ , the weighted Bergman space of D, by $uC_{\varphi }f := u \cdot f \circ \varphi $ for every $f \in A^2_{\alpha }$ . We obtain sufficient conditions for the compactness of $uC_{\varphi }$ in terms of function-theoretic properties of u and $\varphi $ . We also characterize when $uC_{\varphi }$ on $A^2_{\alpha }$ is Hilbert–Schmidt. In particular, the characterization is independent of $\alpha $ when $\varphi $ is an automorphism of D. Furthermore, we investigate the Hilbert–Schmidt difference of two weighted composition operators on $A^2_{\alpha }$ .

In this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Li et al. [A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267(12) (2014), 4753-4774] proved a spectral radius type formula for the approximation numbers of composition operators on analytic Hilbert spaces with radial weights and on $H^{p}$ spaces, $p\geq 1$, involving Green capacity. We prove that their formula holds for a wide class of Banach spaces of analytic functions and weights.

In this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every $n\in \mathbb {N}$ , $n\geq 2$ . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.