On zeros, singular boundary functions, and modules of angular boundary values for one class of functions analytic in a half-plane

Abstract

We obtain the description of the zeros, singular boundary functions, and modules of angular boundary values of the functions $f \neq 0$ which are analytic in the half-plane $C_{+} = \{ z : \Re z > 0 \}$ and satisfy the condition $$( \forall \varepsilon > 0 ) ( \exists c_1 > 0 ) (\forall z \in \mathbb{Ñ}_{+} ): | f ( z ) | \leq c_1 \exp ( (\sigma + \varepsilon) | z \eta ( | z | ) ), $$, where $0 \leq \sigma < +\infty$ is a given number and $\eta$ is a positive function continuously differentiable on $[0; +\infty$ and such that $t\eta'(t)/\eta(t) \rightarrow 0$ as $t \rightarrow + \infty$/