On uniqueness of holomorphic and bounded outside the closed logarithmic sector functions representable by lacunary power series

In the present note it is shown that for a set of positive integers $\Lambda$ a Müntz- type condition holds if and only if there exists a lacunary power series $f(z)=\displaystyle\sum_{v\in\Lambda}f_v/z^v$ that allows an analytic and bounded continuation to the complement of a closed logarithmical sector with vertex at the origin.