We consider polynomials which take integer values on the integers (IVPs), and satisfy an additional growth condition on the natural numbers. Elkies and Speyer, answering a question by Dimitrov, showed that there is a critical exponential growth threshold, such that there are infinitely many IVPs with growth above the threshold and finitely many IVPs below that threshold (of arbitrary degree). In this paper, we give more refined estimates for the number of IVPs satisfying such exponential growth constraints. In addition, we consider a similar problem on the integers, where not necessarily symmetric growth conditions are imposed. Notably, the critical threshold is determined by the logarithmic capacity of an explicit domain.

We explore the injectivity of the evaluation map $\mathrm {eva}_{f,\mathcal {A}} : \mathcal {A}^m \to \mathcal {A} $, where $ \mathcal {A} $ is an associative algebra over a field $ F $ and $ f $ is a polynomial in $ m \geq 1 $ variables with coefficients in F. Our investigation reveals that injectivity is possible only when $ m = 1 $ and $ f $ has degree one; for functions in two or more variables, such injectivity is impossible.

The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis. For example, rational functions enable strikingly effective methods for numerical differentiation and integration; interpolation of equispaced data; locating zeros, poles, and branch points; analytic continuation; computation of inverse functions; imputation of missing data; computation of nonlinear eigenvalues and resonances; model order reduction/reduced order modelling; solution of Wiener–Hopf and Hilbert transform problems; conformal mapping; and solution of the two-dimensional Laplace, biharmonic and Helmholtz equations. The paper also surveys various improvements and generalizations of the AAA algorithm that have developed since its original appearance in 2018.

Let $\Omega _1, \ldots , \Omega _m$ be probability spaces, let ${\mathbf \Omega }=\Omega _1 \times \cdots \times \Omega _m$ be their product and let $A_1, \ldots , A_n \subset {\mathbf \Omega }$ be events. Suppose that each event $A_i$ depends on $r_i$ coordinates of a point $x \in {\mathbf \Omega }$, $x=\left (\xi _1, \ldots , \xi _m\right )$, and that for each event $A_i$ there are $\Delta _i$ other events $A_j$ that depend on some of the coordinates that $A_i$ depends on. Let $\Delta =\max \{5,\ \Delta _i\,:\, i=1, \ldots , n\}$ and let $\mu _i=\min \{r_i,\ \Delta _i+1\}$ for $i=1, \ldots , n$. We prove that if ${\mathbb P}(A_i) \lt (3\Delta )^{-3\mu _i}$ for all $i$, then for any $0 \lt \epsilon \lt 1$, the probability ${\mathbb P}\left ( \bigcap _{i=1}^n \overline {A}_i\right )$ of the intersection of the complements of all $A_i$ can be computed within relative error $\epsilon$ in polynomial time from the probabilities ${\mathbb P}\left (A_{i_1} \cap \ldots \cap A_{i_k}\right )$ of $k$-wise intersections of the events $A_i$ for $k = e^{O(\Delta )} \ln (n/\epsilon )$.

We consider the boundary dynamics of iterated function systems of holomorphic self-maps of the unit disc. Our main result provides a sufficient condition which guarantees that the dynamical behaviour of a left iterated function system in the interior of the unit disc can be extended to the boundary. This generalises an extension of the classical Denjoy–Wolff theorem, due to Bourdon, Matache and Shapiro, to the setting of iterated function systems. To do so, we modify estimates for the Hardy norm of composition operators and combine them with a new technique of perturbing a left iterated function system by elliptic Möbius transformations.

Given a polynomial $\sum _\nu a_\nu X^\nu $ of degree $<d$, bounded by one on the unit disk, how large can $ \left \lvert {a_0+a_1+\ldots +a_n} \right \rvert $ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that n is ‘not too large’ in terms of d. The latter is achieved via a ‘quantitative’ Eneström–Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $ \left \lvert { t_0 a_0 + \ldots + t_{d-1} a_{d-1} } \right \rvert $ for arbitrary complex numbers $t_0,\ldots ,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro’s problem for some choices of n and d.

Let $(Z_{t})_{t \geq 0}$ be a planar Brownian motion running in some domain W and denote by $\tau _{W}$ the exit time of $Z_{t}$ from W. To establish the finiteness of $\mathbf {E}(\sup _{0\leq t\leq \tau _{W}}|Z_{t}|^{p})$ from the finiteness of $\mathbf {E}(|Z_{\tau _{W}}|^{p})$ for some $p>0$, Burkholder [‘Exit times of Brownian motion, harmonic majorization, and Hardy spaces’, Adv. Math. 26(2) (1977), 182–205] imposed an additional condition on the exit time $\tau _{W}$, namely the finiteness of $\mathbf {E}(\log (\tau _{W}))$. Such a condition is typically difficult to verify, since the law of the exit time is often delicate. In this paper, we revisit Burkholder’s condition and propose an alternative viewpoint. Our approach is purely analytic, weaker and formulated in terms of proper analytic maps rather than exit times themselves. This provides a more flexible framework for further applications.

Let $\mathcal {A}$ denote the class of normalised analytic functions f in the open unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$ with $f(0)=0$ and $f'(0)=1$. A function $f\in \mathcal {A}$ is said to be convex if $f(\mathbb {D})$ is convex. We establish a sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of convex univalent (that is, one-to-one) functions in the unit disk $\mathbb {D}$.

In this paper, we investigate the extension of uniformisation results for Gromov hyperbolic spaces beyond the standard geodesic setting. By establishing a Gehring-Hayman type theorem for conformal deformations of any intrinsic Gromov hyperbolic space, we provide a framework for analysing spaces that do not necessarily admit geodesics. As a primary application, we prove that any complete intrinsic hyperbolic space with at least two points in the Gromov boundary can be uniformised by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification between the Gromov boundary of the original space and the metric boundary of the deformed space.

In this article, by utilizing the properties of elliptic functions, we characterize the meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb {C}$, where $L(z)$ is a nonconstant entire function, and m and n are two positive integers. As applications, we also investigate the meromorphic solutions of Fermat-type difference and q-difference equations.

The notion of weighted $\alpha $-composition was introduced by Ruhan Zhao in the 1990s. In this paper, we study several analytic function spaces that are closely related to weighted $\alpha $-composition. These include $\alpha $-Bloch spaces, $F(p,q,s)$ spaces, and Campanato spaces. We obtain derivative-free characterizations for $\alpha $-Bloch spaces and $F(p,q,s)$ spaces, which improve some previous results in the literature. We also obtain a certain version of Carleson measures for Campanato spaces and $F(p,q,s)$ spaces.

From classical trigonometric formulas, complex differential equations of various types have been formulated and widely studied. We investigate perturbed nonlinear complex differential equations and explore their corresponding complex differential systems. Some open questions on nonlinear complex differential equations and systems are proposed.

The well-known proof of Beurling’s Theorem in the Hardy space $H^2$, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces $H^p$ for $0 < p < \infty $ are usually obtained by reduction to the $H^2$ case via inner–outer factorization of $H^p$ functions. In this article, we instead explicitly calculate the metric projection of the unit constant function onto a shift-invariant subspace of the Hardy space $H^p$ when $1<p<\infty $. This problem is equivalent to finding the best approximation in $H^p$ of the conjugate of an inner function. In $H^2$, this approximation is always a constant, but in $H^p$, when $p\neq 2$, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff–James orthogonality and Pythagorean inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-invariant subspaces of $H^p$, as well as the behavior of the zeros of optimal polynomial approximants in $H^p$.

The planar Skorokhod embedding problem was first proposed and solved by Gross [‘A conformal Skorokhod embedding’, Electron. Commun. Probab. 24 (2019), 11 pages; doi:10.1214/19-ECP272]. Gross worked with probability distributions having finite second moment. Boudabra and Markowsky [‘Remarks on Gross’ technique for obtaining a conformal Skorokhod embedding of planar Brownian motion’, Electron. Commun. Probab. 25 (2020), 13 pages; doi:10.1214/20-ECP300] extended the solution to all distributions with a finite pth moment for $p>1$. The case $p=1$ has remained uncovered since then. In this note, we show that the planar Skorokhod embedding problem is solvable for $p=1$ when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$. In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$.

It is known that the condition $|\arg f'(z)|<\pi /2$, $|z|<1$, is not sufficient for an analytic function $f(z)=z+a_2z^2+\cdots $ in $|z|<1$, to be starlike with respect to the origin. We look for the largest $\alpha>0$ such that the condition $|\arg f'(z)|<\alpha \pi /2$ in $|z|<1$ is a sufficient condition for f to be a univalent, starlike, convex, or Bazilevic̆ function.

We consider the family of infinite positive Borel measures $\mu $ in the unit disc, with finite degree of contact at the unit circle, and the set $E_{\mu }$ of points in the unit circle for which every neighborhood contains infinite mass of $\mu $. Assuming that $E_{\mu }$ is a Carleson set, we show that all Blaschke sequences are zero sets for the corresponding Dirichlet space $D_{\mu }$.

In this article, certain classes of homeomorphisms on the complex plane have been considered. Specifically, we investigate ring and lower Q-homeomorphisms with respect to the p-module, as well as homeomorphisms of finite distortion. The behavior at infinity of these mappings is investigated, and sharp estimates for their lower power $\alpha $-order are obtained.

We show that the statement “In every separable pseudometric space there is a maximal non-strictly $\delta $-separated set.” implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and Górka [2]. We also prove several related results.

The primary aim of this paper is to give topological obstructions to Cantor sets in $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus g Cantor set, the first for which the local genus is g at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for $g\geq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus g; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant.