This paper discusses a generalization of Men'shov's theorem which states that a continuous one sheeted complex function \(f\) defined on a domain \(D \subset \mathbb C\), which satisfies the \(K''\) condition at all points of the domain (except at most a countable set), has the property that either \(f\) or \(\bar{f}\) is holomorphic on \(D\). Under an additional condition, on the disposition of rays which appear in the definition of the \(K''\) condition, the author extends the result about the holomorphy of such functions to those that are not necessarily continuous but have the property that \((\log^+| f(z)| )^p\) is integrable with respect to the plane Lebesgue measure for each positive \(p<2\). This is an interesting and self contained paper which defines all the notions which are discussed.