In this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk $ \mathbb {D} $, class of analytic functions $ f $ defined on $ \mathbb {D} $ such that $\mathrm {Re}\left (f(z)\right )<1 $, and class of subordination to a function g in $ \mathbb {D} $. Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.

In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$, the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$, satisfies the inequality $CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where $K(T)$ denotes the Kreiss constant of T and $C>0$ is an absolute constant. He also proved that for $r\ll 1/n,$ the latter bound is asymptotically sharp as $n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit $n\times n$ Toeplitz matrices with arbitrary singleton spectrum $\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to $\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to $\infty $.

The sharp bound for the third Hankel determinant for the coefficients of the inverse function of convex functions is obtained, thus answering a recent conjecture concerning invariance of coefficient functionals for convex functions.

Let ${\mathbb D}$ be the open unit disk, and let $\mathcal {A}(p)$ be the class of functions f that are holomorphic in ${\mathbb D}\backslash \{p\}$ with a simple pole at $z=p\in (0,1)$, and $f'(0)\neq 0$. In this article, we significantly improve lower bounds of the Bloch and the Landau constants for functions in ${\mathcal A}(p)$ which were obtained in Bhowmik and Sen (2023, Monatshefte für Mathematik, 201, 359–373) and conjecture on the exact values of such constants.

We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

Recently, Benini et al showed that, in simply connected wandering domains of entire functions, all pairs of orbits behave in the same way relative to the hyperbolic metric, thus giving us our first insight into the general internal dynamics of such domains. The author proved in a recent paper [G. R. Ferreira. Multiply connected wandering domains of meromorphic functions: internal dynamics andconnectivity. J. Lond. Math. Soc. (2) 106 (2022), 1897–1919] that the same is not true for multiply connected wandering domains, a natural question is how inhomogeneous multiply connected wandering domains can be. We give an answer to this question, in that we show that uniform dynamics inside an open subset of the domain generalizes to the whole wandering domain. As an application of this result, we construct the first example of a meromorphic function with a semi-contracting infinitely connected wandering domain.

In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$. In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$, thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$, where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

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.