Assume a point $z$ lies in the open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and $f$ is an analytic self-map of $\mathbb{D}$ fixing 0. Then Schwarz’s lemma gives $|f(z)|\leq |z|$, and Dieudonné’s lemma asserts that $|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$. We prove a sharp upper bound for $|f^{\prime \prime }(z)|$ depending only on $|z|$.

We describe how to approximate fractal transformations generated by a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:[0,1]\rightarrow [0,1]$ for $i=0,1$. An algorithm is provided for determining the unique parameter value such that the closure of the symbolic attractor $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical. Several examples are given, one in which the $W_{i}$ are affine and two in which the $W_{i}$ are nonlinear. Applications to digital imaging are also discussed.

We give two characterisations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.

We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

We show that positive absolutely norm attaining operators can be characterised by a simple property of their spectra. This result clarifies and simplifies a result of Ramesh. As an application we characterise weighted shift operators which are absolutely norm attaining.

Let $E$ and $D$ be open subsets of $\mathbb{R}^{n+1}$ such that $\overline{D}$ is a compact subset of $E$, and let $v$ be a supertemperature on $E$. We call a temperature $u$ on $D$extendable by$v$ if there is a supertemperature $w$ on $E$ such that $w=u$ on $D$ and $w=v$ on $E\backslash \overline{D}$. Such a temperature need not be a thermic minorant of $v$ on $D$. We show that either there is a unique temperature extendable by $v$, or there are infinitely many. Examples of temperatures extendable by $v$ include the greatest thermic minorant $GM_{v}^{D}$ of $v$ on $D$, and the Perron–Wiener–Brelot solution of the Dirichlet problem $S\!_{v}^{D}$ on $D$ with boundary values the restriction of $v$ to $\unicode[STIX]{x2202}D$. In the case where these two examples are distinct, we give a formula for producing infinitely many more. Clearly $GM_{v}^{D}$ is the greatest extendable thermic minorant, but we also prove that there is a least one, which is not necessarily equal to $S\!_{v}^{D}$.

We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy [‘Generalized notions of amenability’, J. Funct. Anal.  208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied.

A system of functional equations satisfied by the components of a quadratic function is derived via their corresponding circulant matrix. Given such a system of functional equations, general solutions are determined and a stability result for such a system is established.

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $n^{2}/48$ ($n$ even) or $(n+3)^{2}/48$ ($n$ odd). We solve the analogous problem for $m$-gons (for arbitrary but fixed $m\geq 3$) and for polygons (with arbitrary number of sides).

Let $f\in C^{2}(\mathbb{T}^{2})$ have mean value 0 and consider

$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D6FE}\,\text{closed geodesic}}\frac{1}{|\unicode[STIX]{x1D6FE}|}\biggl|\int _{\unicode[STIX]{x1D6FE}}f\,d{\mathcal{H}}^{1}\biggr|,\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}$

$\unicode[STIX]{x1D6FE}:\mathbb{S}^{1}\rightarrow \mathbb{T}^{2}$

$|\unicode[STIX]{x1D6FE}|$

$\unicode[STIX]{x1D6FE}$

$s\geq 2$

$$\begin{eqnarray}|\unicode[STIX]{x1D6FE}|^{s}{\lesssim}_{s}\biggl(\max _{|\unicode[STIX]{x1D6FC}|=s}\Vert \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}f\Vert _{L^{1}(\mathbb{T}^{2})}\biggr)\Vert \unicode[STIX]{x1D6FB}f\Vert _{L^{2}}\Vert f\Vert _{L^{2}}^{-2}.\end{eqnarray}$$

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

Let ${\mathcal{D}}$ be a Schauder decomposition on some Banach space $X$. We prove that if ${\mathcal{D}}$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{T^{n}:n\geq 0\}$ is not $R$-bounded. Likewise, we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to ${\mathcal{D}}$ such that the set $\{e^{-tA}:t\geq 0\}$ is not $R$-bounded.

We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed $n$-disc $\mathbb{D}^{n}$ is the unique topological $n$-manifold (with boundary) whose boundary and interior are homeomorphic to $\mathbb{S}^{n-1}$ and $\mathbb{R}^{n}$, respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to $\mathbb{D}^{3}$.

We prove that positive solutions of an integral equation of Wolff type are radially symmetric and decreasing about some point in $R^{n}$. The hypotheses allow a wider range of exponents and are easier to apply than those in previous work.

The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times \mathbb{R}^{n}$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ respectively.

Let $n$ be a positive integer. A $C^{\ast }$-algebra is said to be $n$-subhomogeneous if all its irreducible representations have dimension at most $n$. We give various approximation properties characterising $n$-subhomogeneous $C^{\ast }$-algebras.