Let $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that

$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$

for all $x\in X$ and $f\in\mathcal{A}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

The characterization by weighted Lipschitz continuity is given for the Bloch space on the unit ball of $\mathbb{R}^n$. Similar results are obtained for little Bloch and Besov spaces.

This paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.

AMS 2000 Mathematics subject classification: Primary 35K55; 35B33