On the growth of the derivative of \(Q_p\) functions (Q2769248)

For \(0<p< \infty\) a function \(f\) analytic in the unit disk \(D\) is in \(Q_p\) when NEWLINE\[NEWLINE\sup_{a\in D}\int \int_D\bigl |f'(z)\bigr |^2 g(z,a)^p dxdy <\inftyNEWLINE\]NEWLINE where \(g(z,a)= \log|(1-\overline a z)/(z-a)|\). Using techniques related to those of \textit{P. B. Kennedy} [Q. J. Math., Oxf. II. Ser. 15, 337-341 (1964; Zbl 0138.05703)] and \textit{D. Girela} [Complex Variables, Theory Appl. 12, 9-15 (1989; Zbl 0696.30031)] the authors prove for \(f\) in \(Q_p\), NEWLINE\[NEWLINE0<p <1,\;\int^1_0(1-r)^p \exp\bigl(2T(r,f') \bigr) dr<\infty,NEWLINE\]NEWLINE and when \(0<p\leq 1\), for almost every \(\theta,|f'(re^{i\theta}) |=o((1-r)^{-(p+1)/2})\), \((r\to 1)\). Each result is shown to be sharp in a sense. The former result is an extension of one of P. B. Kennedy for the Nevanlinna class. A consequence of the latter result concerns the set NEWLINE\[NEWLINEE(f)=\left\{e^{i\theta} \in\partial D\mid V(f, \theta)= \infty \quad\text{where} \quad V(f,\theta)= \int^1_0 \bigl|f'( re^{i \theta}) \bigr|d\theta \right\}:NEWLINE\]NEWLINE if \(f\) is in \(Q_p\) for \(0<p<1\), then \(E(f) \) has linear measure zero. The paper contains an excellent discussion concerning the context for the results and some additional results on the radial growth of the derivative of \(Q_p\) functions.