Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.

The logarithmic coefficients $\unicode[STIX]{x1D6FE}_{n}$ of an analytic and univalent function $f$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalisation $f(0)=0=f^{\prime }(0)-1$ are defined by $\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of $|\unicode[STIX]{x1D6FE}_{n}|$, $n=1,2,3$, for such functions $f$.

We study a representation for the inverse transform of the generalised Fourier–Feynman transform on the function space $C_{a,b}[0,T]$ which is induced by a generalised Brownian motion process. To do this, we define a transform via the concept of the convolution product of functionals on $C_{a,b}[0,T]$. We establish that the composition of these transforms acts like an inverse generalised Fourier–Feynman transform and that the transforms are vector space automorphisms of a vector space ${\mathcal{E}}(C_{a,b}[0,T])$. The vector space ${\mathcal{E}}(C_{a,b}[0,T])$ consists of exponential-type functionals on $C_{a,b}[0,T]$.

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

Let the function $f$ be analytic in $\mathbb{D}=\{z:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. For $0<\unicode[STIX]{x1D6FD}\leq 1$, denote by ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$ the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of $f\in {\mathcal{C}}(\unicode[STIX]{x1D6FD})$, showing that these estimates are the same as those for functions in ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$, thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant $H_{2}=|a_{2}a_{4}-a_{3}^{2}|$, the first three coefficients of $\log (f(z)/z)$ and Fekete–Szegö theorems.

In this paper, we prove some value distribution results which lead to normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family, and improve some recent results. In particular, the main result of this paper leads to a counterexample to the converse of Bloch’s principle.

The aim of this work is to prove the existence of a positive almost periodic solution to a multifinite time delayed nonlinear differential equation that describes the so-called hematopoiesis model. The approach uses the Hilbert projective metric in a cone. With some additional assumptions, we construct a fixed point theorem to prove the desired existence and uniqueness of the solution.

For every $q\in (0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach problem for the class of $q$-convex functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$. In addition, we consider the Fekete–Szegö problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to two conjectures for the class of $q$-starlike functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$.

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

We find all real-valued general solutions $f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution

$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in S$

$S$

$\unicode[STIX]{x1D70E}~:~S\rightarrow S$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in \mathbb{R}^{n}$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$ .

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation

$$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$

$f$

$X$

$Y$

$x,y\in X$

$x+y,x-y\in X$

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$ , given $g$ . There are several approaches involving the use of series expansions of derivatives of $g$ , which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$ ) for the convergence of the series concerned.

Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any $N\in \mathbb{N}$ the entire functions $P$ for which the partition of unity condition $\sum _{\mathbf{n}\in \mathbb{Z}^{d}}P(\mathbf{x}+\mathbf{n})\unicode[STIX]{x1D712}_{[0,N]^{d}}(\mathbf{x}+\mathbf{n})=1$ holds for all $\mathbf{x}\in \mathbb{R}^{d}.$ The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions $(P\unicode[STIX]{x1D712}_{[0,N]^{d}})(\cdot +\mathbf{n})$ can be controlled. In particular, this leads to easy ways of constructing entire functions $P$ such that the functions in the partition of unity belong to the Feichtinger algebra.

In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.