Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.

For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly$\ldots $’, Arnold Math. J. 1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$, where $C(x)$ is a polynomial with $C(w)\ne 0$, and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb {R}_{\mathcal {G}}$ and the reduct of $\mathbb {R}_{\text {an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the gamma function on $(0,\infty )$ and the zeta function on $(1,\infty )$.

This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.

In this paper we introduce and examine the differential subordination of the form

\[ p(z)+zp'(z)\varphi(p(z),zp'(z))\prec h(z),\quad z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}, \]

$h$

$0\in h(\mathbb {D}).$

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.

Let $ \mathcal {B} $ be the class of analytic functions $ f $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C} : |z|<1\} $ such that $ |f(z)|<1 $ for all $ z\in \mathbb {D} $. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ \sum _{n=0}^{\infty }|a_nz^n|\leq 1 $ for $ |z|=r\leq 1/3 $ and $ 1/3 $ cannot be improved. This inequality is called Bohr inequality and the quantity $ 1/3 $ is called Bohr radius. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ |\sum _{n=0}^{N}a_nz^n|<1\;\; \mbox {for}\;\; |z|<{1}/{2} $ and the radius $ 1/2 $ is the best possible for the class $ \mathcal {B} $. This inequality is called Bohr–Rogosinski inequality and the corresponding radius is called Bohr–Rogosinski radius. Let $ \mathcal {H} $ be the class of all complex-valued harmonic functions $ f=h+\bar {g} $ defined on the unit disk $ \mathbb {D} $, where $ h $ and $ g $ are analytic in $ \mathbb {D} $ with the normalization $ h(0)=h^{\prime }(0)-1=0 $ and $ g(0)=0 $. Let $ \mathcal {H}_0=\{f=h+\bar {g}\in \mathcal {H} : g^{\prime }(0)=0\}. $ For $ \alpha \geq 0 $ and $ 0\leq \beta <1 $, let

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha, \beta)=\{f=h+\overline{g}\in\mathcal{H}_{0} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)-\beta\right)>|g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|,\;\; z\in\mathbb{D}\} \end{align*} $$

be a class of close-to-convex harmonic mappings in $ \mathbb {D} $. In this paper, we prove the sharp Bohr–Rogosinski radius for the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $.

For a given Beurling–Carleson subset E of the unit circle $\mathbb {T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from $\mathbb {D}$ to $\mathbb {T}$. The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates $f^n$ grows geometrically fast with the degree $d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of $\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial $f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.

This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

It is well known that for any inner function $\theta $ defined in the unit disk $\mathbb {D}$, the following two conditions: (i) there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim _{n \to \infty } \theta (z) p_n(z) = 1$ for all $z \in \mathbb {D}$ and (ii) $\sup _n \| \theta p_n \|_\infty < \infty $, are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence $\{ \theta (z) p_n(z)\}_n$ as $|z| \to 1$, then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces $K_\theta $. In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in $K_\theta $, and we show that the classical Aleksandrov theorem on density of continuous functions in $K_\theta $ is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces $\mathcal {H}(b)$ and show that the corresponding approximation result also is optimal.

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component could be bounded even if the orbit of the component contains a sequence of annuli whose moduli tend to infinity, and this cannot happen when the maximal modulus of the meromorphic function is uniformly large enough. In this way we extend certain related results for entire functions to meromorphic functions with infinitely many poles.

We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.

In this article, we study a generalized Bohr radius $R_{p, q}(X), p, q\in [1, \infty )$ defined for a complex Banach space X. In particular, we determine the exact value of $R_{p, q}(\mathbb {C})$ for the cases (i) $p, q\in [1, 2]$ , (ii) $p\in (2, \infty ), q\in [1, 2]$ , and (iii) $p, q\in [2, \infty )$ . Moreover, we consider an n-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal {H})$ for an infinite-dimensional complex Hilbert space $\mathcal {H}$ and (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to \infty $ for finite-dimensional X. We also study the multidimensional analog of a related concept called the p-Bohr radius. To be specific, we obtain the asymptotic value of the n-dimensional p-Bohr radius for bounded complex-valued functions, and in the vector-valued case, we provide a lower estimate for the same, which is independent of n.

Given a sequence $\varrho =(r_n)_n\in [0,1)$ tending to $1$, we consider the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $\mathbb {D}$ such that for any compact set K included in the unit circle ${\mathbb {T}}$, different from ${\mathbb {T}}$, the set $\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space ${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in $H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${\mathcal {C}}(K)$ for any compact set $K\subset {\mathbb {T}}$ different from ${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:f\mapsto f(r_n \cdot )$ acting on $H(\mathbb {D})$. Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.

In Gauthier, Manolaki, and Nestoridis (2021, Advances in Mathematics 381, 107649), in order to correct a false Mergelyan-type statement given in Gamelin and Garnett (1969, Transactions of the American Mathematical Society 143, 187–200) on uniform approximation on compact sets K in $\mathbb C^d$ , the authors introduced a natural function algebra $A_D(K)$ which is smaller than the classical one $A(K)$ . In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set K. Finally, we introduce a notion of O-hull of K and strengthen known results.

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.