The sharpness of some cluster set results
Let h(t) be a real-valued function defined for -1\(\leq t\leq 1\) such that (i) h is continuous, (ii) \(h(t)=h(-t)\), (iii) \(h(0)=0\), \(h(1)=1\), \(h(t_ 1)\leq h(t_ 2)\) \((0\leq t_ 1\leq t_ 2\leq 1)\), (iv) h(t)\(\leq t\) and (v) \(h''(t)>0\) \((t\neq 0)\). For a positive constant c, set \(h^ c(x)=h(x/c)\), -c\(\leq x\leq c\). A right h-angle domain in the unit disc \(\Delta\) at \(\tau =e^{i\theta}\in \partial \Delta\) is defined by \[ RA(\theta,a,b,h)=\{re^{i\phi};\quad 1-h^ a(\phi-\theta)< r<1-h^ b(\phi-\theta),\quad 0\leq \phi-\theta \leq a\} \] for two numbers \(0<a<b\). A left h-angle domain LA(\(\theta\),a,b,h) is defined by replacing \(\phi- \theta\) by \(\theta-\phi\) with upper and lower boundary curves by the same inequality. These are called h-angle domains without specifying right or left. Let \[ \Omega (\theta,h)=\{re^{it};\quad 0\leq r\leq 1-h(t- \theta),\quad | t-\theta | \leq 1\}. \] If \(E\subseteq \Delta\) and if \(E\cap \partial \Delta \neq \phi\) then the cluster set of a function defined in \(\Delta\) taking values in the extended complex plane along E will be denoted by C(f,E). Let \(P\subseteq \partial \Delta\). For each \(e^{i\theta}\in \partial \Delta\), let \(\eta(\theta,\epsilon,P)\) be the length of the largest subarc of the arc \((e^{i(\theta - \epsilon)},e^{i(\theta +\epsilon)})\) which does not meet P. The set P is porous at \(e^{i\theta}\) if \[ \overline{\lim}_{\epsilon \to \infty}\frac{\eta (\theta,\epsilon,P)}{\epsilon}>0, \] is porous if it is porous at each \(p\in P\) and is \(\sigma\)-porous if it is the finite or countable union of porous sets. The author shows the sharpness of the following result by the first author [Pac. J. Math. 76, 201-213 (1978; Zbl 0346.30023)]: for all \(e^{i\theta}\in \partial \Delta\), except for a \(\sigma\)-porous set, for any choice of \(0<a<b\) and \(c>0\), \[ C(f,RA(\theta,a,b,h))= C(f,LA(\theta,a,b,h))= C(f,\Omega (\theta,h^ c)), \] and gives the following theorem: Given a \(\sigma\)-porous set \(P\subseteq \partial \Delta\) which can be written as \(P=\cup^{\infty}_{n=1}P_ n\), where \(P_ n\) are closed and porous. There exists a bounded holomorphic function f(z) in \(\Delta\) with the following properties: (i) f(z) is continuous from within \(\Delta\) at each point of \(\partial \Delta -P;\) (ii) For each point \(\tau\in P\), there exist two h-angle domains at \(\tau\) such that the cluster sets of f along these angles are different.
scientific article; zbMATH DE number 4054006