An extension of Bohr's theorem

The following extension of Bohr’s theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane ${\mathbb{C}}_{\theta }=\{s=\sigma +it:\sigma >\theta \}$ that maps ${\mathbb{C}}_{\theta }$ to $\mathbb{C}\setminus \{\alpha ,\beta \}$ for complex numbers $\alpha \mathrm{\ne }\beta$, then $f$ converges uniformly in ${\mathbb{C}}_{\theta +\epsilon }$ for any $\epsilon >0$. The extension is optimal in the sense that the assertion no longer holds should $\mathbb{C}\setminus \{\alpha ,\beta \}$ be replaced with $\mathbb{C}\setminus \{\alpha \}$.