Boundary properties of subclasses of meromorphic functions of

M. M. Dzhrbashyan ; V. S. Zakharyan

Izvestiya. Mathematics , Tome 4 (1970) no. 6, pp. 1273-1354

Résumé

One of the authors [1] has constructed a complete factorization theory for classes of functions meromorphic in the disk $|z|1$. Such a class $N\{\omega\}$ is associated with a given positive continuous function $\omega(x)$ on $[0,1)$ satisfying the conditions $\omega(0)=1$ and $\omega(x)\in L[0,1)$, contains an arbitrary function meromorphic in $|z|1$ for a suitable choice of $\omega(x)$, and coincides in the special case $\omega(x)\equiv1$ with the class $N$ of functions of bounded form of R. Nevanlinna ([2], Chapter VI). In this present paper we study boundary properties of the classes $N\{\omega\}$, which are contained in $N$ when $\omega(x)\uparrow+\infty$ as $x\uparrow1$. We will prove a number of theorems giving various refined metric characteristics of those exceptional sets $E\subset[0{,}2\pi]$ of measure zero on which a function in the class $N\{\omega\}\subset N$ may not possess a radial boundary value. A characteristic of the exceptional sets $E$ will be given in terms of the convex capacity $\operatorname{Cap}\{E;\lambda_n\}$ with respect to a sequence$\{\lambda_n\}$, the Hausdorff $h$-measure $m(E;h)$, or the measure $C_\omega(E)$ associated with the function $\omega(x)$ generating the given class $N\{\omega\}\subset N$.