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4 papers
Boundary properties of subclasses of meromorphic functions of
bounded form
M. M. Dzhrbashyan,
V. S. Zakharyan
Abstract:
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
$N\{\omega\}\subset N$.
UDC:
517.5
MSC: Primary
30A72; Secondary
30A44,
30A68,
30A70,
30A76 Received: 29.05.1970
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