The variation of holomorphic functions on tangential boundary curves (Q1063141)

For \(0\leq \beta \leq 1\), \(D_{\beta}\) denotes the space of holomorphic functions \(f(z)=\sum a_ nz^ n\) in the disc \(| z| <1\) whose coefficients satisfy \[ \sum^{\infty}_{n=1}n^{2(1-\beta)}| a_ n|^ 2<\infty. \] For \(1\leq \gamma <\infty\) and \(c>0\), consider the approach curves \(\Gamma_{\gamma,c}\) defined by \[ \Gamma_{\gamma,c}(r)=r \exp \{ic(1-r)^{1/\gamma}\},\quad 0\leq r<1. \] Let V(f,\(\gamma\),c;t) denote the total variation of f on \(e^{it}\Gamma_{\gamma,c}\). The maximal variation of f is defined by \(MV(f,\gamma,c_ 0;t)=\sup \{V(f,\gamma,c;t): 0<c<c_ 0\}.\) In the paper the author proves the following: 1. If \(\beta \gamma <\), \(f\in D_{\beta}\) and \(c_ 0<\infty\), then \(MV(f,\gamma,c_ 0;t)\) is a bounded function of t on [-\(\pi\),\(\pi\) ]. 2. If \(\leq \beta \gamma <1\), \(f\in D_{\beta}\), \(c_ 0<\infty\), and \(\alpha =2\beta \gamma -1\), then the set of all \(e^{it}\) where \(MV(f,\gamma,c_ 0;t)=\infty\) has \(\alpha\)-capacity 0. 3. If \(\beta \gamma =1\), then there exists an \(f\in D_{\beta}\) that has \(V(f,\gamma,c;t)=\infty\) for every \(c>0\) and every \(t\in [-\pi,\pi]\).