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.

We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power $\operatorname{dist}(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt A1 class of weights in X. To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest E-free holes, these sets characterize the mentioned A1-property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in ${\mathbb R}^n$ that do not belong to $A_\infty.$

If $f=u+iv$ is analytic in the unit disk ${\mathbb D}$, it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if f is a (complex-valued) harmonic function. However, we prove that the same principle holds if we assume, in addition, that f is K-quasiregular in ${\mathbb D}$. The case $0<p<1$ is particularly interesting, and is an extension of the recent Riesz-type theorems for harmonic quasiregular mappings by several authors. Further, we proceed to show that the real and imaginary parts of a harmonic quasiregular mapping have the same degree of smoothness on the boundary.

Motivated by the study of multiplicative linear functionals in reproducing kernel Hilbert space (RKHS) with normalized complete Pick kernel, we define and study the multiplicative linear map between two RKHS. We identify the conditions under which such maps are continuous. Additionally, we prove that any unital cyclicity-preserving linear map is multiplicative. Conversely, we also characterize when a multiplicative linear map is unital cyclicity preserving. These results serve as a generalization of the Gleason–Kahane–Żelazko theorem to the setting of multiplicative maps between two RKHS. We present the composition operator as a natural class of examples of multiplicative linear maps on an RKHS. We also prove that every continuous multiplicative linear operator can be realized as a composition operator on various analytic Hilbert spaces over the unit disc $\mathbb {D}.$

We obtain Wiman–Valiron type inequalities for random entire functions and for random analytic functions on the unit disk that improve a classical result of Erdős and Rényi and recent results of Kuryliak and Skaskiv. Our results are then applied to linear dynamics: we obtain rates of growth, outside some exceptional set, for analytic functions that are frequently hypercyclic for an arbitrary chaotic weighted backward shift.

Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and conjecture a second congruence family by powers of 7, which may be amenable to similar techniques.

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]

$\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$r \leq 4$

$r = 5$

$r \in \{ p-1, p+1, (p+1)/2 \}$

$\Delta_{r/p}$

$J_2(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$J_2(r)$

We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, ɛ-chains are finite collections of points with distance at most ɛ between consecutive points. They play the role of discrete curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter ɛ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradients or of the local Lipschitz constant of Lipschitz functions. The proof of this fact is inspired by a recent technique developed by Eriksson-Bique in Eriksson-Bique (2023 Calc. Var. Partial Differential Equations 62 23). In the possible non-complete setting, we prove that the chain Sobolev space is equal to the one defined via relaxation of the local Lipschitz constant of Lipschitz functions, while in general they are different from the one defined via upper gradients along curves. We apply the theory developed in the paper to prove equivalent formulations of the Poincaré inequality in terms of pointwise estimates involving ɛ-upper gradients, lower bounds on modulus of chains connecting points and size of separating sets measured with the Minkowski content in the non-complete setting. Along the way, we discuss the notion of weak ɛ-upper gradients and asymmetric notions of integral along chains.