Baire category principle and uniqueness theorem

Abstract:

Applying a theorem of Bagemihl and Seidel (1953), we prove that if ${S_2}$ is a set of second category in $(\alpha ,\beta )$, where $0 \leqslant \alpha < \beta \leqslant 2\pi$, and if $f(z)$ is a function meromorphic in the sector $\Delta (\alpha ,\beta ) = \{ z:0 < \left | z \right | < \infty ,\alpha < \arg z < \beta \}$ for which ${\underline {{\operatorname {lim}}} _{r \to \infty }}\left | {f(r{e^{i\theta }})} \right | > 0$, for all $\theta \in {S_2}$, then there exists a sector $\Delta (\alpha ’,\beta ’) \subseteq \Delta (\alpha ,\beta )$ such that $(\alpha ’,\beta ’) \subseteq {\bar S_2},{S_2}$ is second category in $(\alpha ’,\beta ’)$, and $f(z)$ has no zero in $\Delta (\alpha ’,\beta ’)$. Based on this property, we prove several uniqueness theorems.