The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$ , which is closed under complex conjugation and contains $0$ , is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$ . This solves the question posed by K. Mahler completely.

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$ , such that

$$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$

$H_{3,1}(f)$

$$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$. The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.

Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ . We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$ . We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

In this paper we prove uniqueness theorems for mappings $F\in W_{\text{loc}}^{1,n}(\mathbb{B}^{n};\mathbb{R}^{n})$ of finite distortion $1\leq K(x)=\Vert \mathit{DF}(x)\Vert ^{n}/J_{F}(x)$ satisfying some integrability conditions. These types of theorems fundamentally state that if a mapping defined in $\mathbb{B}^{n}$ has the same boundary limit $a$ on a ‘relatively large’ set $E\subset \unicode[STIX]{x2202}\mathbb{B}^{n}$ , then the mapping is constant. Here the size of the set $E$ is measured in terms of its $p$ -capacity or equivalently its Hausdorff dimension.

By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

We find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α

\begin{equation*}\|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p.\end{equation*}

$\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$

We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51(4) (1999), 781–799].

The main aim of this article is to establish analogues of Landau’s theorem for solutions to the $\overline{\unicode[STIX]{x2202}}$ -equation in Dirichlet-type spaces.

In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.

We establish a Schwarz lemma for $V$ -harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.

Let ${\mathcal{H}}ol(B_{d})$ denote the space of holomorphic functions on the unit ball $B_{d}$ of $\mathbb{C}^{d}$ , $d\geq 1$ . Given a log-convex strictly positive weight $w(r)$ on $[0,1)$ , we construct a function $f\in {\mathcal{H}}ol(B_{d})$ such that the standard integral means $M_{p}(f,r)$ and $w(r)$ are equivalent for any $p$ with $0<p\leq \infty$ . We also obtain similar results related to volume integral means.

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$

$\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$

${\mathcal{F}}_{\unicode[STIX]{x1D706}}$

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L 2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence $\ell ^{p}$ spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on $\ell ^{p}$ ’, Sibirsk. Mat. Zh.7 (1966), 146–158].