Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into $\mathbb {R}$-trees have Lipschitz free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into $\mathbb {R}$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to $\ell ^1$, (2) the Lipschitz free space over hyperbolic n-space is isomorphic to the Lipschitz free space over Euclidean n-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an $\mathbb {R}$-tree, has Lipschitz free space isomorphic to $\ell ^1$, and admits a proper, uniformly Lipschitz affine action on $\ell ^1$.

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion. We consider mappings of domains of the Euclidean space that satisfy weighted Poletsky inequality. Assume that, the definition domain is finitely connected on its boundary and, in addition, on the set of all points which are pre-images of the cluster set of this boundary. Then the specified mappings have a continuous boundary extension provided that the majorant in the Poletsky inequality satisfies some integral divergence condition, or has a finite mean oscillation at every boundary point.

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.

In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\mathcal {X}$ of analytic functions on the open unit disc $\mathbb {D}$. We develop a general framework where a method based on a corona theorem can be used to show that if $f,g\in \mathcal {X}$ satisfy $|g(z)|\leq |f(z)|$, for every $z\in \mathbb {D}$, and if g is cyclic, then f is cyclic. We also give sufficient conditions for cyclicity in this context. This enable us to recapture some recent results obtained in de Branges–Rovnayk spaces, in Besov–Dirichlet spaces and in weighted Dirichlet type spaces.

We consider the relationship between the Mahler measure $M(f)$ of a polynomial f and its separation $\operatorname {sep}(f)$. Mahler [‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262] proved that if $f(x) \in \mathbb {Z}[x]$ is separable of degree n, then $\operatorname {sep}(f) \gg _n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relationship and led to questions about the optimal exponent on $M(f)$. However, there has been relatively little study concerning upper bounds on $\operatorname {sep}(f)$ in terms of $M(f)$. We prove that if $f(x) \in \mathbb {C}[x]$ has degree n, then $\operatorname {sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$; for example, if it has only real roots.

Let ${\mathcal {E}}$ be a complex elliptic curve and S be a non-empty finite subset of ${\mathcal {E}}$. We show that the functions $\tilde {\Gamma }$ introduced in [BDDT] out of string theory motivations give rise to a basis (as a vector space) of the minimal algebra $A_{{\mathcal {E}}{\smallsetminus } S}$ of holomorphic multivalued functions on ${\mathcal {E}}{\smallsetminus } S$ which is stable under integration, introduced in [EZ]; this basis is alternative to the basis of $A_{{\mathcal {E}}{\smallsetminus } S}$ constructed in loc. cit. using elliptic analogs of the hyperlogarithm functions.

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

Let $2\leq p<\infty $ and X be a complex infinite-dimensional Banach space. It is proved that if X is p-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space $\mathscr {H}^{\text {weak}}_p(X)$ to the Hardy space $\mathscr {H}^+_p(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood–Paley inequality for Dirichlet series is established.

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt’s theorem on nearly invariant subspaces of the backward shift operator on $H^2(\mathbb {D})$ as well as its many generalizations to the setting of de Branges spaces.

We show the existence of transcendental entire functions $f: \mathbb {C} \rightarrow \mathbb {C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of f has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of f, answering a question of Rippon and Stallard [Eremenko points and the structure of the escaping set. Trans. Amer. Math. Soc. 372(5) (2019), 3083–3111]. Our proof relies on a quasiconformal-surgery approach developed by Burkart and Lazebnik [Interpolation of power mappings. Rev. Mat. Iberoam. 39(3) (2023), 1181–1200].

In this article, by the use of nth derivative characterization, we obtain several some sufficient conditions for all solutions of the complex linear differential equation

$$ \begin{align*}f^{(n)}+A_{n-1}(z)f^{(n-1)}+\ldots+A_1(z)f'+A_0(z)f=A_n(z) \end{align*} $$

$A_i(z) (i=0,1,\ldots ,n)$

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}({\mathbb R}^n;{\mathbb R}^n)$-mappings. Then we show on the plane that this relaxed definition can be used to prove Sobolev regularity, and that these ‘finely quasiconformal’ mappings are in fact quasiconformal.

In this note, it is shown that the differential polynomial of the form $Q(f)^{(k)}-p$ has infinitely many zeros and particularly $Q(f)^{(k)}$ has infinitely many fixed points for any positive integer k, where f is a transcendental meromorphic function, p is a nonzero polynomial and Q is a polynomial with coefficients in the field of small functions of f. The results are traced back to Problems 1.19 and 1.20 in the book of research problems by Hayman and Lingham [Research Problems in Function Theory, Springer, 2019]. As a consequence, we give an affirmative answer to an extended problem on the zero distribution of $(f^n)'-p$, proposed by Chiang and considered by Bergweiler [Bull. Hong Kong Math. Soc. 1(1997), p. 97–101].

In this article, we study the action of the the Hilbert matrix operator $H$ from the space of bounded analytic functions into conformally invariant Banach spaces. In particular, we describe the norm of $H$ from $H^\infty $ into $\text {BMOA}$ and we characterize the positive Borel measures $\mu $ such that $H$ is bounded from $H^\infty $ into the conformally invariant Dirichlet space $M(D_\mu )$. For particular measures $\mu $, we also provide the norm of $H$ from $H^\infty $ into $M(D_\mu )$.

Let µ be a finite positive Borelmeasure on $[0,1)$ and $\alpha \gt -1$. The generalized integral operator of Hilbert type $\mathcal {I}_{\mu_{\alpha+1}}$ is defined on the spaces $H(\mathbb{D})$ of analytic functions in the unit disc $\mathbb{D}$ as follows:

\begin{equation*}\mathcal {I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t),\ \ f\in H(\mathbb{D}),\ \ z\in \mathbb{D} .\end{equation*}

In this paper, we give a unified characterization of the measures µ for which the operator $\mathcal {I}_{\mu_{\alpha+1}}$ is bounded from the Bloch space to a Bergman space for all $\alpha \gt -1$. Additionally, we also investigate the action of $\mathcal {I}_{\mu_{\alpha+1}}$ from the Bloch space to the Hardy spaces and the Besov spaces.

In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator

\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}

acting on a number of important analytic function spaces on $\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces $A^{p,\lambda}$ and Dirichlet–Morrey spaces $\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.

We present a one-parameter family Fλ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of Fλ. Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and $\infty$. The family Fλ is the simplest in this class, and its parameter space shows open sets of λ-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.

We consider uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems, we consider spectral data depending partially or fully on the spectrum, derivative values at the spectrum, Clark measure, or the spectrum of the negative of a meromorphic inner function. Moreover, we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of the Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions.

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also generalizes and improves many previous second main theorems for holomorphic maps from ${\mathbb{C}}$ intersecting hypersurfaces (moving and fixed) in projective varieties.

For $r\in(0,1)$, let $\mu \left( r\right) $ be the modulus of the plane Grötzsch ring $\mathbb{B}^2\setminus[0,r]$, where $\mathbb{B}^2$ is the unit disk. In this paper, we prove that

\begin{equation*}\mu \left( r\right) =\ln \frac{4}{r}-\sum_{n=1}^{\infty }\frac{\theta _{n}}{2n}r^{2n},\end{equation*}

with $\theta _{n}\in \left( 0,1\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\mu \left( r\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\mu \left( r\right) $ are established.